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

Linear Equations in Two Variables

An equation of the form \( ax+by+c=0 \) with a and b not both zero is a linear equation in two variables; it has infinitely many solutions, each a point on a straight line. If \( b=0 \) the line is parallel to the y-axis, if \( a=0 \) it is parallel to the x-axis, and if \( c=0 \) it passes through the origin.
To fix both unknowns we need a pair of equations, \( a_1x+b_1y+c_1=0 \) and \( a_2x+b_2y+c_2=0 \).

Methods of Solving

Substitution: express one variable from an equation and put it in the other. Elimination: multiply the equations so that a variable has equal coefficients, then add or subtract. Cross multiplication: \( \frac{x}{b_1c_2-b_2c_1}=\frac{y}{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1} \), usable when \( a_1b_2-a_2b_1\neq 0 \).
Equations such as \( \frac{a}{x}+\frac{b}{y}=c \) become linear after the substitution \( u=\frac1x,\ v=\frac1y \); remember to go back to x and y at the end.

Nature of the Solution

Compare the ratios of the coefficients. If \( \frac{a_1}{a_2}\neq\frac{b_1}{b_2} \) the lines intersect at one point: a unique solution and a consistent pair. If \( \frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2} \) the lines are parallel: no solution, an inconsistent pair. If \( \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \) the lines coincide: infinitely many solutions.
Questions asking for a value of k use exactly these conditions.

Graphical Solution

Draw each line from two or three points, such as its intercepts with the axes. The coordinates of the common point are the solution. The lines together with the axes often enclose a triangle, whose area is half base times height with the base taken along an axis.

Word Problems

Choose letters for the two unknowns, translate each statement into an equation and solve. Typical types: numbers and fractions, digits of a number (a two-digit number is \( 10a+b \); the reversed number is \( 10b+a \)), ages, speed with stream or wind (downstream speed \( x+y \), upstream \( x-y \)), work, coins and notes, cost and profit percentage.
Always check the answer against both original conditions.