Pair of Linear Equations in Two Variables is a chapter in the CBSE Class 10 Mathematics syllabus from Mathematics. This chapter hub brings together revision notes, practice questions, worksheets, flashcards, formula sheet to help students learn, practice, and revise Pair of Linear Equations in Two Variables effectively.

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Pair of Linear Equations in Two Variables

NCERT Class 10 Mathematics Chapter 3: Pair of Linear Equations in Two Variables (Pages 24–37)

Class 10 CBSE hub Mathematics chapters Mathematics

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Summary of Pair of Linear Equations in Two Variables

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Pair of Linear Equations in Two Variables at a Glance

Board

CBSE

Class

Class 10

Subject

Mathematics

Book

Mathematics

Chapter

Pages

24–37

Resources

7 study resources

Pair of Linear Equations in Two Variables Summary

In this chapter, students explore pairs of linear equations in two variables, learning how to represent real-life situations with mathematical equations. It begins with an engaging context, illustrating how to set up equations using scenarios involving simple activities, such as spending money at a fair. The importance of understanding linear equations is emphasized, as they are foundational in representing relationships between two variables. The chapter introduces key concepts such as consistent and inconsistent pairs of equations. Students learn that a consistent pair of equations has at least one solution, while an inconsistent pair has none. Furthermore, within consistent equations, students are exposed to dependent equations, which have infinitely many solutions. The graphical method is a focal point, where students learn to visualize equations on a graph. They understand that the graphs of these equations can represent three distinct scenarios: a unique intersection point indicating a single solution, parallel lines showing no solutions, and coincident lines reflecting multiple solutions. Practical examples illustrate how to determine the nature of the relationships between the equations. Through hands-on exercises, students practice graphing equations and interpreting the results. For example, they explore how to determine whether two equations are consistent or inconsistent by analyzing their graphical representation. In addition to the graphical method, the chapter highlights algebraic methods for solving these equations, providing students with tools for finding solutions when graphical methods prove cumbersome. The emphasis is on understanding that equations can be manipulated and rearranged to reveal their solutions, broadening students’ problem-solving skills. Exercises reinforce the concepts learned, allowing students to practice forming pairs of equations from word problems and determining their solutions graphically. Key themes throughout the chapter include the importance of visualizing equations, understanding their relationships, and the practical applications of linear equations in everyday life. By the end of the chapter, students are equipped with the necessary skills to confidently approach and solve problems involving pairs of linear equations.

Pair of Linear Equations in Two Variables Revision Guide

Download the Pair of Linear Equations in Two Variables revision guide with key points, summaries, and quick revision notes for CBSE Class 10 Mathematics.

Key Points

Definition of Linear Equation in Two Variables.

An equation in the form ax + by + c = 0 where a, b, and c are constants.

Concept of Consistent and Inconsistent Equations.

Consistent equations have at least one solution; inconsistent have none.

Types of Solution Sets: Unique, Infinite.

Unique solution: lines intersect; infinite solutions: lines overlap.

The Graphical Method of Solution.

Solutions are found by graphing equations and analyzing intersections.

Algebraic Methods: Substitution.

Substitute one variable from one equation into the other to find solutions.

Algebraic Methods: Elimination.

Add or subtract equations to eliminate one variable and solve for the other.

Standard Form of Linear Equations.

Form occurs when equations are rearranged to ax + by = c for clarity.

Slope-Intercept Form Explanation.

Can be written as y = mx + c where m is the slope and c is the y-intercept.

Identifying Parallel Lines.

Lines are parallel if their slopes are equal and do not intersect.

Identifying Coincident Lines.

Lines are coincident if they lie on top of each other: every point is shared.

Finding Intersections Algebraically.

Set equations equal to each other to find intersection points explicitly.

Application Example in Real Life.

Budget problems involving costs of activities can be modeled using equations.

Understanding the Coefficients.

Coefficients represent the rate at which y changes relative to x in equations.

Conditions for Consistency.

For consistency, the ratio \(rac{a1}{a2}=rac{b1}{b2}\) must be equal when solutions are infinite.

Using Graphs to Verify Solutions.

Plotting graphs can be a visual aid to confirm the accuracy of algebraic solutions.

Parameters of Equations.

Changing parameters affects the graph shape and can shift intersection points.

Interpreting Equation Solutions.

Solutions provide intersection points, indicating where equations balance.

Special Cases: Vertical Lines.

Vertical lines represent undefined slopes; have equations of the form x = a.

Examples of Inconsistent Systems.

Examples include parallel lines that never intersect, implying no solutions.

Complexity: Non-Integer Solutions.

Solutions can be fractions or decimals, necessitating accurate graph readings.

Comparison of Coefficients for Solutions.

By analyzing \(rac{a1}{a2}\), \(rac{b1}{b2}\), \(rac{c1}{c2}\), the solution type can be determined.

Pair of Linear Equations in Two Variables Practice Questions & Answers

Practice important questions and exam-style problems from Pair of Linear Equations in Two Variables. These questions cover key topics from the CBSE Class 10 Mathematics syllabus.

How to practice: Start with the questions below to test your understanding of Pair of Linear Equations in Two Variables. Use the revision guide to review concepts you find difficult, then come back and retry the questions for better retention.

What is the general form of a linear equation in two variables?

Single Answer MCQ

Q-00173690

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If a linear equation is given by 3x + 4y = 12, what is the value of y when x = 0?

Single Answer MCQ

Q-00173691

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In the equation 2x - 3y = 6, what is the slope of the line?

Single Answer MCQ

Q-00173692

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Which of the following pairs (x, y) satisfies the equation 5x + 2y = 10?

Single Answer MCQ

Q-00173693

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If the solution to a linear equation is (x, y) = (3, 2), which of the following equations is satisfied?

Single Answer MCQ

Q-00173694

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What is the graphical representation of the equation y = 2x + 1?

Single Answer MCQ

Q-00173695

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What type of linear equation pair has no solutions?

Single Answer MCQ

Q-00173696

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Identify the type of solution for the equations 2x + 4y = 8 and x + 2y = 4.

Single Answer MCQ

Q-00173697

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View all 221 Pair of Linear Equations in Two Variables questions

Which pair of linear equations is consistent?

What does the point of intersection of two linear equations represent?

If the pair of equations are dependent, what is their solution?

If 3x + 5y = 15 and 2x - y = 4 are given, what is the common solution of these equations?

Determine if the equations x + y = 5 and 2x + 2y = 10 are consistent or inconsistent.

From the equation y = -3x + 6, what is the y-intercept?

How can you determine if a pair of linear equations is inconsistent?

How can you determine the number of solutions to a system of linear equations?

What does it mean for two equations to be equivalent?

What would the system of equations 2x + 3y = 6 and 4x + 6y = 12 represent?

If the equations 3x + 2y = 6 and 6x + 4y = 12 are given, what type of pairing do they represent?

For the equations 4x + y = 8 and 2x + y = 3, which relationship shows they are inconsistent?

For the equations 2y = 3x + 6 and 4y = 6x + 12, what is their relationship?

How can you express the line 5x - 4y + 8 = 0 in slope-intercept form?

Which of the following equations represent a pair of lines that are parallel?

Which outcome indicates a pair of equations is consistent?

What indicates that the pairs of equations 4x - y = 2 and 2x - 0.5y = 1 are equivalent?

If the equations y = 2x + 3 and y = 2x - 1 are given, what is their relationship?

What type of system has no solution?

If two lines intersect at a point, how many solutions do the equations have?

Which conditions indicate that two lines are coincident?

What can be inferred if two equations have infinitely many solutions?

Which pair of equations represents an inconsistent system?

Which graphical interpretation matches a consistent pair of equations?

How many solutions does the system of equations represent if they are dependent?

What changes in a system of equations when they become inconsistent?

Which of the following is a true statement about lines representing dependent equations?

If the lines represented by two equations are perfectly overlapping, which of the following is true?

Which of the following correctly describes the scenario of having two parallel lines?

In terms of solutions, what can be inferred from a pair of equations if their graphs are two intersecting lines?

If y = 3x + 1, find the value of y when x = 2.

If 2x + 3y = 12 and y = 4, what is the value of x?

Solve the equations: y = 2x + 1 and y = -x + 4 using substitution method.

In the equations x + y = 10 and y = x + 2, find the value of x.

Using substitution, solve the system of equations: x + 2y = 10 and x - y = 1.

If y = -2x + 5 and 4x + 2y = 10, find x.

Solve for y: 6x + 3y = 9 and x = 1.

If y = 3 and x + y = 7, what is the value of x?

Given 3x + 2y = 14 and 2y = 28 - 3x, find y.

If x + y = 15 and y = 2x - 4, what is the value of y?

From the equations 4x + 7y = 1 and 2x - 3y = 3, find the value of x.

If y = -5 and 3x + 7y = -28, find x.

Determine x from the equation: 1/2x + y = 3 and y = 3 - 1/2x.

Consider the system of equations: x - y = 4 and y = x + 5, solve for both variables.

From the equations: 5x - 2y = 5 and 3y = 15 - 2x, determine the value of y.

What type of pair of linear equations has no solution?

Which ratios must be compared to determine if a pair of linear equations is consistent?

If the equations 2x + 3y = 6 and 4x + 6y = 12 represent a dependent pair, what can be said about their solutions?

For the equations 3x + 5y = 15 and 6x + 10y = 30, what is their relationship?

When a pair of linear equations intersects at a single point, they are categorized as?

Which pair of equations is inconsistent?

When evaluating the equations 5x – 4y + 8 = 0 and 9x + 3y + 12 = 0, what does a ratio comparison of coefficients indicate?

Identify the consistent pair of equations:

If two equations result in the same line when graphed, they are classified as?

Which equation represents a dependent pair?

Given 2x - y = 3 and 4x - 2y = 6, what type of system is this?

What is the solution for the equations x + y = 5 and 2x + 2y = 10?

If both equations have the same slope and different intercepts, what type of lines do they represent?

Which of the following pairs is inconclusive?

What can be concluded if the equations 3x + 2y = 5 and 2x – 3y = 7 are combined and yield no solutions?

What can be concluded if two lines are parallel?

To determine if the system has a unique solution, which condition must hold?

Determine the slopes of the equations 2x + 3y = 6 and 4x + 6y = 12.

In which case do the equations 2x + y = 4 and 4x + 2y = 8 represent an inconsistent system?

Find the value of y in the equation 3x + 6y = 18 when x = 1.

To solve a pair of consistent equations graphically, what is needed?

Which scenario describes consistent and dependent equations?

What is the slope of the line represented by the equation 3x + 2y = 6?

What is the graphical representation of the equations x + 3y = 6 and 3x + 9y = 18?

What is the determinant condition for inconsistent equations?

How many solutions do two parallel lines have?

For the pair of equations 5x + 10y = 20 and 3x + 6y = 12, what type of system is it?

Solve for y in the equations 4x + 5y = 20 when x = 2.

If the equations are 2x + 3y = 6 and 4x + 6y = 12, what type of pairing is it?

What is the first step in solving the system of equations 2x + 3y = 6 and 4x - y = 5 using the substitution method?

Which method is useful to solve the equations 3x + 4y = 18 and 2x - y = 1 when both variables appear in both equations?

If the equations 5x + 2y = 20 and 5x + y = 14 are solved, what will be the value of y?

When using the elimination method, which operation can be most useful for aligning equations?

In the equation 3x - 2y = 7, how can you express x in terms of y?

What is the solution of the system of equations when 2x + 3y = 12 and x = 2?

What represents a unique solution in a system of linear equations?

Which of the following pairs of equations have no solution?

If the equations 7x + 5y = 10 and 14x + 10y = 20 are given, what can be said about their solution?

What is the result of solving the system 2x + 5y = 20 and 4x + 10y = 40?

What does the consistent system of linear equations imply?

In solving x + y = 10 and 2x + 3y = 20 using the elimination method, what should you do first?

In the equations 8x - 4y = 12 and -2x + y = 3, what is the value of y when x = 0?

What is the value of x when solving the equations 5x + 2y = 16 and y = 4?

What is the first step in applying the elimination method for the equations 2x + 3y = 12 and 4x + 6y = 24?

For the system of equations: 3x + 4y = 10 and 6x + 8y = 20, which method can be used to determine if they have unique solutions?

In the system of equations 2x + 3y = 6 and 2x + 3y = 12, what can you determine about the solutions?

When using the elimination method, what is the purpose of multiplying equations by certain factors?

Solve for x and y in the equations 4x + 5y = 9 and 2x + 3y = 6 using the elimination method.

What is the elimination method used for in solving linear equations?

Given the equations 5x + 2y = 10 and 10x + 4y = 20, what type of solutions do they represent?

When two equations are parallel, what can be said about their slopes?

Which system of equations can be solved using the elimination method: 3x + 4y = 5 or 6x + 8y = 15?

If the equations are 3x + 2y = 12 and 9x + 6y = 36, how would you classify their solutions?

To eliminate y in the equations 7x + 2y = 14 and 5x - 2y = 6, what operation should be performed?

When forming a system of linear equations, which characteristics determine their solution type?

What happens to the system of equations if you multiply one of the equations by a non-zero constant?

Identify the word that indicates that the cosmetics industry is anticipated to grow.

The sum of numerator and denominator of a fraction is 4 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes 1/3. Find the fraction.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. Representing the above situation, frame two linear equations in two variables.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. Solve the equations and hence find the numbers.

Solve the following equations graphically: x + y = 7 and 2x - 5y = 7.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. Show that the equations have unique solution.

If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has:

Two sections, A and B, of class X contributed a total of ₹1500 for the Uttarakhand flood victims. The contribution from X-A was ₹100 less than that of X-B. Graphically, find the amounts contributed by both sections.

Assertion (A): The value of p for which the system of equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0 is consistent is 4. Reason (R): The system of equations a1x + b1y = c1 and a2x + b2y = c2 is consistent with infinitely many solutions, if a1/a2 = b1/b2 = c1/c2.

Solve the following system of equations for x and y: x/2 + 2y/3 = -1 and x - y/3 = 3.

x and y are complementary angles such that x : y = 1 : 2. Express the given information as a system of linear equations in two variables and hence solve it.

Solve the following system of equations graphically: x + 3y = 6; 2x - 3y = 12.

Assertion (A): The system of linear equations 3x − 5y + 7 = 0 and −6x + 10y + 14 = 0 is inconsistent. Reason (R): When two linear equations don’t have unique solution, they always represent parallel lines.

Use graphical method to solve the system of linear equations: x = −3 and 5x − 2y = −5.

If the pair of linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is consistent and dependent, then

Aarush bought 2 pencils and 3 chocolates for ₹11 and Tanish bought 1 pencil and 2 chocolates for ₹7 from the same shop. Represent this situation in the form of a pair of linear equations. Find the price of 1 pencil and 1 chocolate, graphically.

The pair of linear equations 3x/2 + 5y/3 = 7 and 9x + 10y = 14 is:

If the pair of linear equations: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is consistent and dependent, then

Represent the following pair of linear equations graphically and hence comment on the condition of consistency of this pair: x - 5y = 6; 2x - 10y = 12.

Equation of another line parallel to the line represented by 2x − 6y = 7 is:

Solve the following system of equations graphically: x − 2y = 3, 3x − 8y = 7.

Five years ago, Adil was thrice as old as Bharat. Ten years later, Adil shall be twice as old as Bharat. Form the linear equations representing the above information.

For the age problem of Adil and Bharat, show that the system of equations is consistent with unique solution.

Find the present ages of Adil and Bharat.

Equation of another line parallel to the line represented by 2x – 6y = 7 is:

Solve the following pair of linear equations by substitution method: 2x + 3y = 5, –3x + y = –2.

The sum of the digits of a two-digit number is 11. The number obtained by reversing the digits is 9 more than the original number. Formulate the pair of linear equations for this situation.

Show that the pair of linear equations obtained from the given two-digit number situation is consistent.

Solve graphically the pair of linear equations obtained from the two-digit number situation and find the number.

Equation of a line coincident with 2.5x - 2y = 3 is:

Five years ago, Adil was thrice as old as Bharat. Ten years later Adil shall be twice as old as Bharat. To know the present ages of Adil and Bharat, form the linear equations representing the above information.

Solve the following system of equations graphically: x - 2y = 3, 3x - 8y = 7.

Five years ago, Adil was thrice as old as Bharat. Ten years later Adil shall be twice as old as Bharat. Show that the system of equations is consistent with unique solution.

Five years ago, Adil was thrice as old as Bharat. Ten years later Adil shall be twice as old as Bharat. Find the present ages of Adil and Bharat.

If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has:

Determine graphically, the coordinates of vertices of a triangle whose equations are 2x − 3y + 6 = 0, 2x + 3y − 18 = 0 and x = 0. Also, find the area of this triangle.

If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equations has:

Draw the graph of the pair of linear equations x – y + 2 = 0 and 4x – y – 4 = 0. Calculate the area of the triangle formed by the lines so drawn and the x-axis.

Assertion (A): The value of p for which the system of equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0 is consistent is 4. Reason (R): The system of equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ is consistent with infinitely many solutions, if a₁/a₂ = b₁/b₂ = c₁/c₂.

Solve for x and y: 0.1x + 0.3y = 1; 0.2x – 0.1y = –0.1.

Solve the following system of equations graphically: x + 3y = 6; 2x – 3y = 12.

x and y are complementary angles such that x : y = 1 : 2. Express the given information as a system of linear equations in two variables and hence solve it.

Solve for x and y: 3x + 5y = 8; 5x - 3y = 2.

Solve the following system of equations graphically: x + 3y = 6; 2x - 3y = 12.

x and y are complementary angles such that x : y = 1 : 2. Express the given information as a system of linear equations in two variables and hence solve it.

If (0, 0) is the solution of the equation x + y = c − 1, then the value of c is:

Solve for x and y: 3x + 2y = 65; 2x + 3y = 60.

Find the value of c for which the following pair of linear equations has infinitely many solutions: cx + 3y = c − 3; 12x + cy = c.

An academy offering cricket coaching bought 10 bats and 5 balls for ₹32,500. Later, the academy bought 2 bats and 8 balls for ₹10,000. If there is no change in the cost of the bat and of the ball, find the cost of 1 bat and 1 ball.

Solve graphically the following pair of linear equations: 2x − y = 2 and 4x − y = 4. Also, write the coordinates of the points where the lines represented by these equations cut the y-axis.

The number of solutions of the system of equations x = 3, y = -1 is:

Find the value of c for which the following pair of linear equations has infinitely many solutions: cx + 3y = c - 3; 12x + cy = c.

Solve for x and y: 3x + 2y = 65; 2x + 3y = 60.

Solve graphically the following pair of linear equations: 2x - y = 2 and 4x - y = 4. Also, write the coordinates of the points where the lines represented by these equations cut the y-axis.

The number of solutions of the system of equations x = a, x = b (a ≠ b) is:

Find the value of c for which the following pair of linear equations has infinitely many solutions: cx + 3y = c − 3; 12x + cy = c.

Solve for x and y: 3x + 2y = 65; 2x + 3y = 60.

Solve graphically the following pair of linear equations: 2x − y = 2 and 4x − y = 4. Also, write the coordinates of the points where the lines represented by these equations cut the y-axis.

For what value(s) of k does the system of equations kx + 2y = 3 and 2x + y = 5 have a unique solution?

Solve the following system of equations graphically: 2x – 3y = –6 and x = 3.

A fraction becomes 1/3 when 1 is subtracted from its numerator and it becomes 1/4 when 8 is added to its denominator. Find the fraction.

Find the value of k for which the following pair of linear equations will have infinitely many solutions: kx + 3y – (k – 3) = 0 and 12x + ky – k = 0. Hence, find any two solutions of the given pair of equations.

For what value(s) of k, is the system of equations kx + 2y = 3 and 2x + y = 5 inconsistent?

Solve the following system of equations graphically: x + 2y = 10 and y = 3.

A fraction becomes 1/3 when 1 is subtracted from its numerator and it becomes 1/4 when 8 is added to its denominator. Find the fraction.

The system of linear equations given by x = a and y = b is:

Solve the following system of linear equations graphically: x + y = 5 and x − y = 3.

A fraction becomes 1/3, when 1 is subtracted from the numerator and it becomes 1/4, when 8 is added to its denominator. Find the fraction.

Find the value of k for which the following pair of linear equations will have infinitely many solutions: kx + 3y − (k − 3) = 0 and 12x + ky − k = 0. Hence, find any two solutions of the given pair of equations.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. Representing the above situation, frame two linear equations in two variables.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. Show that the equations have unique solution.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. Solve the equations and hence find the numbers.

Solve the following equations graphically: x + y = 7 and 2x - 5y = 7.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. (i) Represent the situation by two linear equations in two variables. (ii) Show that the equations have unique solution. (iii) Solve the equations and hence find the numbers.

Solve the following equations graphically: x + y = 7 and 2x - 5y = 7.

Seema spent 15 minutes on an exercise bicycle and 30 minutes on a double cross walker and burned 435 calories. She spent 30 minutes on the exercise bicycle and 40 minutes on the double cross walker and burned 690 calories. Represent the situation in terms of a pair of linear equations in two variables.

Seema spent 15 minutes on an exercise bicycle and 30 minutes on a double cross walker and burned 435 calories. She spent 30 minutes on the exercise bicycle and 40 minutes on the double cross walker and burned 690 calories. Solve both equations to find the values of the variables using elimination method.

Seema spent 15 minutes on an exercise bicycle and 30 minutes on a double cross walker and burned 435 calories. She spent 30 minutes on the exercise bicycle and 40 minutes on the double cross walker and burned 690 calories. Solve both equations to find the values of the variables using substitution method.

Seema spent 15 minutes on an exercise bicycle and 30 minutes on a double cross walker and burned 435 calories. She spent 30 minutes on the exercise bicycle and 40 minutes on the double cross walker and burned 690 calories. Solve both equations to find the values of the variables using elimination method.

Seema spent 15 minutes on an exercise bicycle and 30 minutes on a double cross walker and burned 435 calories. She spent 30 minutes on the exercise bicycle and 40 minutes on the double cross walker and burned 690 calories. Solve both equations to find the values of the variables using substitution method.

Seema daily goes to a park to exercise on machines available there. When Seema spent 15 minutes on exercise bicycle and 30 minutes on double cross walker, she received a message of burning 435 calories. When she spent 30 minutes on exercise bicycle and 40 minutes on double cross walker, she received a message of burning 690 calories. Represent the above situation in terms of a pair of linear equations in two variables.

Seema daily goes to a park to exercise on machines available there. When Seema spent 15 minutes on exercise bicycle and 30 minutes on double cross walker, she received a message of burning 435 calories. When she spent 30 minutes on exercise bicycle and 40 minutes on double cross walker, she received a message of burning 690 calories. Solve both equations to find the values of the variables using elimination method.

Seema daily goes to a park to exercise on machines available there. When Seema spent 15 minutes on exercise bicycle and 30 minutes on double cross walker, she received a message of burning 435 calories. When she spent 30 minutes on exercise bicycle and 40 minutes on double cross walker, she received a message of burning 690 calories. Solve both equations to find the values of the variables using substitution method.

The value of k for which the pair of linear equations kx – 3y = 5, 4x – 6y = 10 has infinitely many solutions, is:

Solve the following pair of linear equations: 31x + 43y – 117 = 0; 43x + 31y = 105.

When 1 is subtracted from the numerator and 2 is added to the denominator of a fraction, it becomes 1/2. When 7 is subtracted from the numerator and 2 is subtracted from the denominator, the fraction becomes 1/3. Find the fraction.

Assertion (A): The system of linear equations 3x - 5y + 7 = 0 and -6x + 10y + 14 = 0 is inconsistent. Reason (R): When two linear equations don’t have unique solution, they always represent parallel lines.

Use graphical method to solve the system of linear equations: x = -3 and 5x - 2y = -5.

Assertion (A): The system of linear equations 3x – 5y + 7 = 0 and –6x + 10y + 14 = 0 is inconsistent. Reason (R): When two linear equations don’t have unique solution, they always represent parallel lines.

Use graphical method to solve the system of linear equations: y = –3 and x + 2y = 4.

Solve the system of linear equations: x = 4 and 3x - 2y = 6 graphically.

The value of k for which the system of linear equations kx – y – 2 = 0 and 6x – 2y – 3 = 0 has infinitely many solutions, is

In a class test, Veer scored 6 more than twice as many marks as Kevin scored. If one of them had scored 4 more marks, their total score would have been 40. Find the marks obtained by Veer and Kevin.

Solve the linear equations 3x + y = 14 and y = 2 graphically.

The value of k for which the system of linear equations x/2 + y/3 = 5 and 2x + ky = 7 is inconsistent, is

In a class test, Veer scored 6 more than twice as many marks as Kevin scored. If one of them had scored 4 more marks, their total score would have been 40. Find the marks obtained by Veer and Kevin.

Solve the linear equations 3x + y = 14 and y = 2 graphically.

The value of k for which the system of linear equations x/2 + y/3 = 5 and 2x + ky = 7 is inconsistent, is

In a class test, Veer scored 6 more than twice as many marks as Kevin scored. If one of them had scored 4 more marks, their total score would have been 40. Find the marks obtained by Veer and Kevin.

Solve the linear equations 3x + y = 14 and y = 2 graphically.

If x = 1 and y = 2 is a solution of the pair of linear equations 2x − 3y + a = 0 and 2x + 3y − b = 0, then:

Vijay invested certain amounts of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. He received ₹1,860 as the total annual interest. However, had he interchanged the amounts of investments in the two schemes, he would have received ₹20 more as annual interest. How much money did he invest in each scheme?

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Q-00208138

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Pair of Linear Equations in Two Variables Practice Worksheets

Download and practice Pair of Linear Equations in Two Variables worksheets to improve problem-solving accuracy and speed for CBSE Class 10 Mathematics exams.

Pair of Linear Equations in Two Variables - Practice Worksheet

This worksheet covers essential long-answer questions to help you build confidence in Pair of Linear Equations in Two Variables from Mathematic for Class 10 (Mathematics).

Practice

Questions

Define a pair of linear equations in two variables. Give an example and explain the significance of each component in the equations.

A pair of linear equations in two variables is a set of equations that each describes a straight line. The general form is ax + by + c = 0, where a, b, and c are constants. For example, 2x + 3y - 6 = 0 and x - y + 1 = 0 are two linear equations. Here, 'a' and 'b' are the coefficients of variables x and y, 'c' is the constant term, and 'x' and 'y' are the variables. Each equation represents a line on the Cartesian plane, and the solution is the point where these lines intersect.

Explain the graphical method of solving a pair of linear equations. Provide an example with a step-by-step solution.

The graphical method involves drawing the lines represented by each equation on a graph. The solution is found at the point of intersection of these lines. For example, to solve the equations y = 2x + 1 and y = -x + 3 graphically, plot points for both equations using various values of x. Draw the lines and identify the intersection point at (2, 5). This point is the solution of the equations since it satisfies both equations.

What are consistent and inconsistent pairs of linear equations? Provide conditions for both and examples.

Consistent pairs of linear equations have at least one solution, while inconsistent pairs have no solutions. Conditions for consistency include: 1) The lines intersect at a single point (unique solution), or 2) The lines are coincident (infinitely many solutions). An example of consistent equations is x + y = 4 and 2x + 2y = 8. An example of inconsistent equations is x + y = 5 and x + y = 3, where the lines are parallel.

Using algebraic methods, solve the pair of equations: 2x + 3y = 12 and x - y = 1. Show all steps.

To solve using substitution, rearrange the second equation for x: x = y + 1. Substitute in the first equation: 2(y + 1) + 3y = 12. Expand and simplify: 2y + 2 + 3y = 12, leading to 5y = 10. Thus, y = 2. Substitute back to find x: x = 2 + 1 = 3. Therefore, the solution is (3, 2).

Demonstrate how to determine if two lines represented by linear equations are coincident or parallel by finding an appropriate pair of equations and analyzing their coefficients.

Consider the equations 4x + 5y - 10 = 0 and 2x + rac{5}{2}y - 5 = 0. Rearranging gives them similar forms. Compare ratios of coefficients: a1/a2 = 4/2 = 2; b1/b2 = 5/(5/2) = 2; c1/c2 = -10/-5 = 2. Since all ratios are equal, these equations represent coincident lines and have infinitely many solutions.

Write a real-life problem that can be modeled using a pair of linear equations and solve it.

A baker makes cupcakes and cookies. The number of cupcakes he makes is twice the number of cookies. If he sells these for a total of ₹150, where cupcakes are ₹5 each and cookies are ₹3 each, we let x be the number of cookies and y be cupcakes. This gives us the equations y = 2x and 5y + 3x = 150. Substituting the first equation into the second yields 5(2x) + 3x = 150, simplifying to 10x + 3x = 150. Thus, 13x = 150, leading to x = 11.54 (approx. 11 cookies), and y = 2*11 = 22 cupcakes.

Explain what it means for a pair of equations to have infinitely many solutions and provide examples illustrating this condition.

A pair of equations has infinitely many solutions if they represent the same line - they are dependent. For example, the equations 2x + 4y = 8 and x + 2y = 4 are equivalent; multiplying the second equation by 2 yields the first. Thus, every solution of one is also a solution of the other. Graphically, they overlay perfectly.

Formulate the equations based on the following conditions: a total of 30 birds, where the number of geese is double the number of ducks, and solve the equations.

Let x be the number of ducks and y be the number of geese. The equations are y = 2x and x + y = 30. Substituting the first equation into the second gives x + 2x = 30, thus 3x = 30 leading to x = 10. Hence, y = 20, indicating 10 ducks and 20 geese.

Provide a method for verifying the solution of a pair of linear equations after finding it.

To verify a solution, substitute the values of the variables back into the original equations. For example, if a solution for the equations 2x + 3y = 6 and x - y = 2 is x = 3 and y = 0, substituting gives 2(3) + 3(0) = 6 and 3 - 0 = 2. Both check out, confirming the solution is correct.

Pair of Linear Equations in Two Variables - Mastery Worksheet

This worksheet challenges you with deeper, multi-concept long-answer questions from Pair of Linear Equations in Two Variables to prepare for higher-weightage questions in Class 10.

Mastery

Questions

Akhila spent ₹20 on rides and games where each ride costs ₹3 and each game costs ₹4. If the number of Hoopla games she played is half the number of rides, formulate the linear equations and solve them graphically. Show the graphical representation and the intersecting point.

Let x be the number of rides and y be the number of Hoopla games. The equations will be y = 0.5x and 3x + 4y = 20. Graph the equations to find the intersection point (4, 2) where Akhila had 4 rides and played 2 games.

Champa purchased skirts and pants. She stated that the number of skirts is two less than twice the number of pants, while also being four less than four times the number of pants. Formulate the equations and find the number of pants and skirts she bought.

Let x = number of pants and y = number of skirts. The equations are y = 2x - 2 and y = 4x - 4. Solving these gives x = 1 and y = 0 (1 pant, 0 skirts).

A shop sold 5 pencils and 7 pens for ₹50 and 7 pencils and 5 pens for ₹46. Derive the equations, solve them using substitution or elimination method, and justify your solution.

Let x = cost of one pencil and y = cost of one pen. The equations are 5x + 7y = 50 and 7x + 5y = 46. Using substitution method, derive costs x = 2 and y = 3.

Evaluate if the pair of equations 2x + 3y = 8 and 4x + 6y = 16 are consistent or inconsistent. Provide a comparative analysis of their coefficients.

The ratios are a1/a2 = 0.5, b1/b2 = 0.5, c1/c2 = 0.5, indicating dependent lines (coincident) hence infinitely many solutions.

Discover if the equations x - 2y = 3 and 2x - 4y = 6 are coincident, parallel, or intersecting. Justify through graphical or algebraic methods.

These are parallel since their ratios yield inconsistency between c1 and c2, leading to no solution.

For a rectangular garden with a perimeter of 72 meters, where the length is 10 m more than its width, formulate the equations and determine the dimensions graphically.

Let width = w and length = l. The equations 2l + 2w = 72 and l = w + 10 lead to dimensions: width = 16 m, length = 26 m.

If the area of a rectangle is given by the equation l*w = 90 with length 5 more than width, create an equation and find dimensions of the rectangle.

Let w = width, l = w + 5. Then, w(w + 5) = 90 leads to solving a quadratic equation with dimensions found w = 6, l = 11.

Graphically represent the pair of equations 3x + 2y = 12 and x - 4y = -8. Identify the solution point and the slope of each line.

Graphing shows intersection at (2, 3) with slopes calculated as -3/2 and 1/4 respectively.

Formulate and solve the equations for two lines that are coincident based on the general form ax + by + c = 0. Provide a practical scenario.

Example equations: x + 2y - 4 = 0 and 2x + 4y - 8 = 0; both represent the same line with infinite solutions.

Determine if the equations 5x - 6y + 15 = 0 and 10x - 12y + 30 = 0 are consistent or inconsistent by comparing coefficients.

Since the second is a multiple of the first, they represent the same line, thus having infinitely many solutions.

Pair of Linear Equations in Two Variables - Challenge Worksheet

The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Pair of Linear Equations in Two Variables in Class 10.

Challenge

Questions

How can the concept of inconsistency in pairs of linear equations be applied to predict outcomes in real-life situations? Provide an example and analyze the impacts of such inconsistencies.

Explore situations like budget constraints where offered solutions are unachievable. Assess implications of inconsistency on decisions.

Critique the graphical and algebraic methods used to solve pairs of linear equations. Discuss the advantages and disadvantages of each method with relevant examples.

Assess how each method applies to various kinds of linear equations and determines solution types. Provide clear instances for comparison.

Given the equations x - y = 4 and 2x + 2y = 16. Analyze the conditions that define their intersection or parallelism. What could this mean in a practical context?

Graphically analyze and determine the nature of these lines and what real-world scenarios they apply to, incorporating case interpretations.

Propose a pair of linear equations based on a scenario where multiple solutions exist. Determine how you would derive the equations and discuss their implications.

Design equations that depict concurrent relationships, demonstrating how to identify infinitely many solutions.

Examine how a change in one variable affects the outcome in the context of linear equations. Provide a specific example with calculations.

Analyze the equations and showcase how variable modification leads to unique or multiple solutions.

Create a real-life problem that can be represented by a pair of inconsistent equations. Show how the conclusions drawn would affect decision-making in that scenario.

Elucidate on real-life implications of inconsistent equations within the problem and its potential outcomes.

Discuss the role of parameterization in finding solutions for pairs of linear equations, using examples. How does it help simplify complex equations?

Provide examples showing parameterization benefits in deriving solutions.

Interpret a pair of linear equations that represent a pair of overlapping geographic regions. How would adjustments to one equation affect the overall area represented?

Analyze the set equations and the implications of modification on the represented area.

Formulate a pair of linear equations illustrating a scenario in sport team selection and analyze potential outcomes based on team compositions.

Create the equations based on player abilities or roles, explaining possible team dynamics resulting from solutions.

Devise a hypothetical scenario where changing one linear equation's coefficients drastically alters the solution set. Evaluate the importance of each coefficient.

Analyze the modifications and their impact on solutions, stressing coefficient roles in solution existence.

Pair of Linear Equations in Two Variables Formula Sheet

Use this Class 10 Mathematics Pair of Linear Equations in Two Variables Formula Sheet for quick revision before school exams and CBSE exams. It brings together the important formulas, key concepts, and worked examples in one place so students can revise faster and download a printable PDF for offline study.

Important Formulas

y = mx + c

y is the dependent variable, m is the slope (change in y/change in x), x is the independent variable, and c is the y-intercept (value of y when x=0). This equation represents a line in the Cartesian plane.

Slope (m) = (y₂ - y₁) / (x₂ - x₁)

This formula calculates the slope between two points (x₁, y₁) and (x₂, y₂). A steeper slope indicates a steeper line on the graph.

x₁/a₁ = x₂/a₂ = y₁/b₁ = y₂/b₂

This expresses the condition under which two lines are either parallel or coincident. Here, (x₁, y₁) and (x₂, y₂) are coordinates of points on lines with slopes a₁ and a₂ respectively.

y - y₁ = m(x - x₁)

This point-slope form of the equation of a line allows you to write the equation of a line given a point (x₁, y₁) and the slope m.

General form: ax + by + c = 0

This is the standard form of a linear equation in two variables, where a, b, and c are real numbers. This form is often used for solving systems of equations.

x + y = k

This represents a line where the sum of x and y is constant (k). Useful for easily finding intercepts.

Elimination Method: a₁x + b₁y = c₁ and a₂x + b₂y = c₂

Utilizes addition or subtraction to eliminate one variable and solve for the other, making it efficient for finding solutions of linear equations.

Substitution Method: Solve for x or y in one equation, and substitute into the other.

This method replaces one variable with an expression from another equation, simplifying the system for easier solving.

Graphical Method: Plot the equations on a graph.

Visualize the solutions by plotting both equations. The intersection point(s) represent the solution(s) of the equations.

Infinite solutions criterion: If a₁/a₂ = b₁/b₂ = c₁/c₂

Determines that the lines are coincident (same line), thus having infinitely many solutions.

Worked Examples

3x + 4y = 20

This equation represents a linear relationship between the variables x and y. It can be solved using various methods for specific solutions.

2x + 3y = 9

A linear equation representing another line in the same two-dimensional space. Finding solutions involves intersection with another line.

x – 2y = 0

This equation can help derive the relationship between x and y where y is directly proportional to x, ideal for linear relationships.

y = (1/2)x

This equation indicates that y is half of x, easily demonstrates proportionality and linearity.

x + 2y = 8

This equation describes a line where the sum of x and twice y equals 8, useful in graphical interpretations.

5x – 3y = 12

A linear equation that can be used to find specific values of x and y through methodical solving.

2x - 5 = y

This rearranged form shows y in terms of x, allowing for direct calculation of y values from given x.

x + y = 6

Indicates a line where the sum of x and y is constant, important in algebraic applications.

4x + y = 24

Another standard linear equation form to find relationships between x and y across a range of values.

y = 3x + 2

This slope-intercept form indicates that the line crosses the y-axis at 2 with a slope of 3, illustrating direction.

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Pair of Linear Equations in Two Variables Frequently Asked Questions

Explore the chapter on Pair of Linear Equations in Two Variables for Class 10, covering key concepts, solving methods, and real-life applications.

QWhat are the applications of linear equations in daily life?

Linear equations are used in various real-life scenarios such as budgeting, planning, and resource allocation. For example, if you spend a certain amount on items and have a budget limit, you can use linear equations to determine how many items you can buy without exceeding your budget. Understanding these equations helps in making informed decisions.

QWhat is the graphical method of solving linear equations?

The graphical method involves plotting the equations of the linear pair on a graph. The point at which the lines intersect represents the solution to the equations. If the lines coincide, there are infinitely many solutions; if they are parallel, there is no solution, indicating inconsistent equations.

QHow do you determine if a pair of linear equations is consistent or inconsistent?

To determine consistency, you can compare the slopes of the equations. If the lines intersect (unique solution), the equations are consistent. If they are parallel (no solution), they are inconsistent. Coincident lines indicate infinitely many solutions, confirming that the equations are consistent.

QWhat is the substitution method for solving linear equations?

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This helps simplify the problem to a single equation with one variable, making it easier to find the values of both variables step by step.

QCan you explain the elimination method of solving linear equations?

The elimination method involves adding or subtracting the equations to eliminate one variable, simplifying the system to a single equation. Once one variable is found, it can be substituted back to find the other variable. This method is especially useful for equations that are easily additive or subtractive.

QWhat does it mean for two lines to be parallel?

Two lines are parallel if they have the same slope but different intercepts, indicating that they will never intersect. In the context of linear equations, this means that the system of equations has no solution, thus classified as an inconsistent pair.

QHow do you represent a real-life situation using linear equations?

To represent a real-life situation with linear equations, identify the variables involved, translate the relationships into mathematical expressions, and create equations. For example, if a person buys items with constraints on amount spent, these can be modeled as linear equations based on cost and quantity.

QWhat is meant by dependent linear equations?

Dependent linear equations are those that represent the same line when graphed, resulting in infinitely many common solutions. Such equations can be derived from each other by scaling, indicating that they describe the same relationship between variables.

QWhat are the types of linear equations?

Linear equations can be classified into three types: consistent with a unique solution (intersecting lines), inconsistent with no solution (parallel lines), and dependent with infinitely many solutions (coincident lines). Understanding these classifications helps in analyzing their graphical representations.

QHow do you use the elimination method to solve specific equations?

To use the elimination method, align the equations and strategically multiply them if necessary to ensure that adding or subtracting the equations will eliminate one variable. Solve for that variable, then substitute back to find the other variable.

QWhat role does slope play in understanding linear equations?

The slope represents the steepness and direction of a line in a linear equation. It helps determine whether two lines are parallel, intersect, or coincide. In the context of equations, the slope also informs how changes in one variable affect the other, illustrating relationships quantitatively.

QHow can you find the solution to a system of linear equations graphically?

To find the solution graphically, graph each equation on the same coordinate plane. The coordinates of the point where the lines intersect represent the solution of the system. If the lines do not intersect, the system may be inconsistent or dependent.

QWhat is the significance of the constants in a linear equation?

The constants in a linear equation (the intercepts) define where the line intersects the axes. They provide information about the starting point of a linear relationship and help in graphing the equation accurately.

QCan all linear equations be solved using both graphical and algebraic methods?

Yes, all linear equations can be approached using both graphical and algebraic methods. While the graphical method provides a visual representation, the algebraic methods allow for precise calculations. The choice depends on the specific context and preference for a particular problem.

QWhat impacts do different coefficients have in linear equations?

The coefficients of the variables in linear equations determine the slope and position of the line. Changing these values can shift the line's steepness and its location on the graph, affecting the relationship represented by the equations.

QIn what cases do linear equations have no solution?

Linear equations have no solution when they represent parallel lines, meaning they have the same slope but different y-intercepts. This indicates that the equations are inconsistent and do not share any points in common.

QHow do you graphically represent a consistent pair of equations?

A consistent pair of equations can be graphically represented by intersecting lines. The point of intersection denotes the unique solution to the equations, highlighting where both variables satisfy the conditions set by the equations.

QWhat is the difference between dependent and independent equations?

Dependent equations are those that yield the same line and thus have infinitely many solutions, while independent equations yield distinct lines that intersect at a single point, resulting in a unique solution. Understanding this is crucial in classifying linear equation systems.

QWhat is meant by the term 'unique solution' in the context of linear equations?

A unique solution refers to a single pair of values for the variables that satisfies both equations in a system. This occurs when the lines representing the equations intersect at exactly one point on a graph.

QWhy is it important to understand the properties of linear equations?

Understanding the properties of linear equations is essential for solving mathematical problems across various fields, including economics, physics, and engineering. It equips students with tools to model real-world scenarios and analyze relationships between variables effectively.

QHow can one transform a linear equation for easier solving?

Transforming a linear equation might involve rearranging it to isolate one variable or adjusting coefficients to facilitate the elimination or substitution methods. This often simplifies the equations and enhances the efficiency of finding solutions.

QHow does understanding variables in equations enhance problem-solving skills?

Understanding variables allows students to translate real-world problems into mathematical expressions. This skill is key in applying mathematical concepts to solve complex situations in academics and everyday life.

QWhat methods are most effective for teaching linear equations?

Effective methods for teaching linear equations include visual aids such as graphs, interactive activities that involve real-world applications, and step-by-step examples that reinforce both graphical and algebraic techniques for finding solutions.

QHow can practice with linear equations improve overall mathematical fluency?

Regular practice with linear equations enhances students' understanding of algebraic principles and improves their ability to manipulate and solve equations. This fluency is essential for tackling higher-level mathematics and applying these skills in practical contexts.

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Pair of Linear Equations in Two Variables Flashcards

Revise key terms and definitions from Pair of Linear Equations in Two Variables with interactive flashcards. Quick recall practice for CBSE Class 10 Mathematics.

These flash cards cover important concepts from Pair of Linear Equations in Two Variables in Mathematics for Class 10 (Mathematics).

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What is a pair of linear equations?

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A pair of linear equations consists of two equations that relate two variables, often represented as ax + by + c = 0.

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What is a consistent pair of linear equations?

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A consistent pair of linear equations has at least one solution, meaning the lines intersect.

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What is an inconsistent pair of linear equations?

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An inconsistent pair has no solution, meaning the lines are parallel and never intersect.

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What is a dependent pair of linear equations?

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A dependent pair has infinitely many solutions, represented by coincident lines.

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How are linear equations graphically represented?

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Linear equations are represented as lines on the Cartesian plane, with their intersections indicating solutions.

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When do two lines have a unique solution?

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When the lines intersect at exactly one point, the pair is consistent with a unique solution.

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How do you determine if lines intersect or are parallel?

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Use the ratios a1/a2, b1/b2, c1/c2: If they are equal, lines are parallel or coincident; if not, they intersect.

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Example showing graphical solution?

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For equations x + 3y = 6 and 2x - 3y = 12, the intersection shows the solution (6,0).

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What is the standard form of a linear equation?

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The standard form is expressed as Ax + By + C = 0, where A, B, C are constants.

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What is the elimination method?

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A technique where one variable is eliminated by adding or subtracting equations to simplify solving.

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What is the substitution method?

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A technique where one equation is solved for one variable, and then substituted into the other equation.

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How do you apply equations in word problems?

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Translate the words into equations, like for costs: 3x + 4y = total spent and y = 1/2x.

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What is a common mistake in solving equations?

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Confusing the signs during addition/subtraction of equations can lead to incorrect solutions.

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What does the intersection point of two lines represent?

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The intersection point (x, y) is the solution to the pair of linear equations.

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What are coincident lines?

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Coincident lines are lines that lie on top of each other, representing infinitely many solutions.

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What defines parallel lines in equations?

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Parallel lines have the same slope but different y-intercepts, indicating no intersection.

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How do you plot linear equations?

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Find two or more points that satisfy the equations and draw lines through them.

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How to simplify equations before solving?

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You can divide equations by common factors to make calculations easier.

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What is a system of equations?

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A system is a set of equations with the same variables that can be solved together.

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