CBSE Class 10 Mathematics Pair of Linear Equations in Two Variables Notes Set 09

Pair of Linear Equations in Two Variables

A linear equation in two variables has the form ax + by + c = 0 with a and b not both zero. Its graph is a straight line, and every point on the line gives a solution. Two such equations, a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, form a pair, and a solution of the pair satisfies both.

Nature of Solutions

Compare the ratios of the coefficients.
1. If a₁/a₂ ≠ b₁/b₂, the lines intersect at one point: exactly one solution (consistent, independent).
2. If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines coincide: infinitely many solutions (consistent, dependent).
3. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel: no solution (inconsistent).

Methods of Solving

Graphical method: draw both lines and read the intersection point.
Substitution method: express one variable from one equation and substitute in the other.
Elimination method: multiply the equations by suitable numbers so that one variable has equal coefficients, then add or subtract.
Cross-multiplication: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁).

Equations Reducible to Linear Form

Pairs such as 12/(x − 1) + 40/(x + 1) = 8 become linear after substituting p = 1/(x − 1) and q = 1/(x + 1). Solve for p and q, then return to x. Equations with moduli are solved by splitting into cases.

Word Problems

Choose two variables for the two unknown quantities, translate each condition into an equation, solve, and check against the story. Typical types are digit problems (number = 10a + b), age problems, work problems (rates add), money and interest, mixtures, and boat-and-stream problems (speed downstream = b + s, upstream = b − s).

Worked Example

Solve x − y = 1 and 2x + y = 8.
Adding gives 3x = 9, so x = 3 and y = 2. The lines meet at (3, 2). With the y-intercepts (0, −1) and (0, 8), the triangle they form with the y-axis has area ½ × 9 × 3 = 13.5 square units.