Question

Question 1 (ii)
Which of the following pairs of linear equations has a unique solution, no solution or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.
2x + y = 5 ; 3x +2y =8

Answer 1

2x + y = 5
3x +2y = 8

Compare the given equations with

$a_{1}x+b_{1}y+c_1=0and{a}_{2}x+b_{2}y+c_2=0$.

$\frac{a_1}{a_2}=\frac{2}{3}$
$\frac{b_1}{b_2}=\frac{1}{2}$
$\frac{c_1}{c_2}=\frac{-5}{-8}=\frac{5}{8}$
$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
Therefore, they will intersect each other at a unique point and thus, there will be a unique solution for these equations. By cross-multiplication method,
$\frac{x}{b_1{c}_2-b_2{c}_1}=\frac{y}{c_1{a}_2-c_2{a}_1}=\frac{1}{a_1{b}_2-a_2{b}_1}$
$\frac{x}{-8-(-10)}=\frac{y}{-15+16}=\frac{1}{4-3}$
$\frac{x}{2}=1,\frac{y}{1}=1$
$\therefore x=2,y=1$