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Formula Sheet
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Pair of Linear Equations in Two Variables – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the 'Pair of Linear Equations in Two Variables' chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
General Form: a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0
Represents a pair of linear equations in two variables x and y. a₁, a₂ are coefficients of x; b₁, b₂ are coefficients of y; c₁, c₂ are constants. Used to model real-world problems involving two variables.
Condition for Consistency: a₁/a₂ ≠ b₁/b₂
If the ratio of coefficients of x and y are not equal, the equations are consistent and have a unique solution. Essential for determining the nature of solutions without solving.
Condition for Inconsistency: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
If the ratios of coefficients are equal but not equal to the ratio of constants, the equations are inconsistent and have no solution. Helps in identifying parallel lines graphically.
Condition for Dependency: a₁/a₂ = b₁/b₂ = c₁/c₂
If all ratios are equal, the equations are dependent and have infinitely many solutions. Indicates coincident lines on a graph.
Substitution Method: y = (c₁ - a₁x)/b₁
Express y in terms of x from one equation and substitute into the other to find x. Useful when one variable can be easily isolated.
Elimination Method: a₁x + b₁y = c₁, a₂x + b₂y = c₂
Multiply equations to make coefficients of one variable equal and subtract to eliminate that variable. Efficient for solving systems quickly.
Cross-Multiplication Method: (x/(b₁c₂ - b₂c₁)) = (y/(c₁a₂ - c₂a₁)) = (1/(a₁b₂ - a₂b₁))
Direct formula to find x and y. Remember the pattern for quick application in exams.
Graphical Solution: Intersection point of two lines
The point where the two lines represented by the equations intersect is the solution. Visual method to understand the nature of solutions.
Sum of two numbers: x + y = S
S represents the sum of two numbers x and y. Basic form to model problems involving sum of quantities.
Difference of two numbers: x - y = D
D represents the difference between two numbers x and y. Useful in problems involving age differences or quantity comparisons.
Equations
Example 1: x + y = 14, x - y = 4
Simple pair to find two numbers whose sum is 14 and difference is 4. Solve by addition or substitution method.
Example 2: 2x + 3y = 9, 4x + 6y = 18
Represents dependent equations with infinitely many solutions. Ratios a₁/a₂ = b₁/b₂ = c₁/c₂.
Example 3: x + 2y = 4, 2x + 4y = 12
Inconsistent pair with no solution. Ratios a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
Age Problem: s - 7 = 7(t - 7), s + 3 = 3(t + 3)
Models age relations. s and t represent current ages. Solve to find specific ages.
Cost Problem: 2x + 3y = 9, 4x + 6y = 18
Models cost of items. Shows dependency when costs are proportional.
Digit Problem: x + y = 6, x - y = 2
Finds digits of a number. x is tens digit, y is units digit. Solve to find the number.
Perimeter Problem: l = w + 4, (l + w) = 36
l is length, w is width. Models rectangular garden dimensions. Solve to find l and w.
Fraction Problem: (x + 2)/(y + 2) = 9/11, (x + 3)/(y + 3) = 5/6
Finds original fraction x/y. Solve using substitution or cross-multiplication.
Income-Expenditure: 9x - 4y = 2000, 7x - 3y = 2000
Models monthly incomes and expenditures. Solve to find x and y representing incomes.
Rail Problem: x + 2y = 4, 2x + 4y = 12
Inconsistent system representing parallel rails that never meet. No solution exists.
Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
