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title: "Pair of Linear Equations in Two Variables"
board: "CBSE"
curriculum: "CBSE"
class: "Class 10"
subject: "Mathematics"
book: "Mathematics"
chapter: "Pair of Linear Equations in Two Variables"
chapter_slug: "pair-of-linear-equations-in-two-variables"
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---

# Pair of Linear Equations in Two Variables
This chapter discusses how to represent real-life situations using linear equations in two variables, and explores various methods to solve them, including graphical and algebraic techniques.

---

## Knowledge Snapshot
| Field | Details |
| :--- | :--- |
| Class | Class 10 |
| Subject | Mathematics |
| Book | Mathematics |
| Chapter | Pair of Linear Equations in Two Variables |
| Pages | 24-37 |

---

## Chapter Summary
### Short Summary
The chapter introduces linear equations in two variables and explains how to find their solutions through different methods including graphical and algebraic approaches.

### Detailed Summary
The chapter begins with an example of Akhila at a fair where she engages in activities involving rides and games, expressed as a pair of linear equations. It describes the graphical method for finding solutions of linear equations, categorizing pairs as consistent or inconsistent based on their graphical representation (intersection, parallelism, or coincidence). Further, it explains the comparison of coefficients method and provides illustrative examples for graphical and algebraic solving methods. Various exercises encourage application of these concepts.

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## Topic-Wise Explanation
### Introduction to Linear Equations
Linear equations are introduced through a practical example involving Akhila's spending at a fair, leading to the formation of equations to represent her rides and games. The equations are:
$$y = \frac{1}{2} x$$
$$3x + 4y = 20$$

### Graphical Method of Solution of a Pair of Linear Equations
This section explains how to determine the nature of solutions to linear equations by graphing:
- Unique solution (intersecting lines)
- No solution (parallel lines)
- Infinitely many solutions (coincident lines)

### Types of Linear Equations
Pairs of linear equations can be classified based on their solutions:
- Inconsistent: no solution
- Consistent: at least one solution (unique or infinite)
- Dependent: infinitely many solutions.

### Examples of Linear Equations
Examples illustrate consistent and inconsistent pairs, showcasing their graphical representation.

### Algebraic Methods of Solving Linear Equations
This part covers algebraic techniques for solving linear equations, especially useful for non-integral coordinate solutions where graphical methods may be impractical.

### Substitution Method
The substitution method involves isolating one variable in terms of another and substituting into the second equation.

### Elimination Method
The elimination method focuses on eliminating one variable to solve the equations. It can be done via addition or subtraction of the equations.

### Summary of Linear Equations
The chapter concludes with a summary of key concepts, the graphical and algebraic methods, and the characteristics of the types of linear equations.

---

## Core Ideas
| Idea | Explanation |
| :--- | :--- |
| Consistent Pair | A pair of linear equations has at least one solution. |
| Inconsistent Pair | A pair of linear equations has no solutions. |
| Dependent Pair | A pair of linear equations has infinitely many solutions. |

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## Key Concepts
| Concept | Meaning |
| :--- | :--- |
| Linear Equation | An equation of the first degree involving two variables. |
| Graphical Solution | Finding the intersection point of two lines on a graph. |

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## Important Points for Revision
* Linear equations can be represented graphically.
* Pairs of linear equations can be consistent or inconsistent.
* Different methods exist for solving linear equations, including substitution and elimination.
* The graphical method helps visualize the nature of the solutions.
* Comparison of coefficients can help classify pairs as intersecting, coincident, or parallel.

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## Practice Questions
### Short Answer Questions
1. Form the pair of linear equations for a scenario where ten students participated in a quiz, with four more girls than boys.
2. Solve graphically the equations 5 pencils + 7 pens = ₹50 and 7 pencils + 5 pens = ₹46.
3. Determine if the equations 5x – 4y + 8 = 0 and 9x + 3y + 12 = 0 intersect.
4. Compare the ratios for the equations 3x + 2y = 5 and 2x – 3y = 7 to determine consistency.
5. Find dimensions of a garden with half the perimeter equal to 36 m, where the length is 4 m more than the width.

### Long Answer Questions
1. Illustrate graphically the equations x+y=5 and 2x+2y=10.
2. Verify and solve the equations x-2y=0 and 3x+4y=20 graphically.
3. Given the equations 2x + 3y - 8 = 0, provide an equation that leads to parallel lines.

---

## Related Concepts
* Dependent and Independent Variables
* Algebraic Methods of Solving Equations

---

## Source Attribution
| Field | Value |
| :--- | :--- |
| Source | Edzy |
| Reference Type | examSubjectBookChapter |
| Reference ID | 69f8685f293c3123114d5722 |
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