Question

On comparing the ratios $\frac{a_1}{a_2},\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent.
(i) $3x+2y=5;2x-3y=7$

(ii) $2x-3y=8;4x-6y=9$

(iii) $\frac{3}{2}x+\frac{5}{3}y=7;9x-10y=14$

(iv) $5x-3y=11;-10x+6y=-22$

(v) $\frac{4}{3}x+2y=8;2x+3y=12$

Answer 1

(i) $\frac{a_1}{a_2}=\frac{3}{2},\frac{b_1}{b_2}=\frac{2}{-3},\frac{c_1}{c_2}=\frac{5}{7}$

$\because \frac{a_1}{a_2}\ne \frac{b_1}{b_2}\ne \frac{c_1}{c_2}$,

the lines intersect and have an unique consistent solution.

(ii) $\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\frac{b_1}{b_2}=\frac{-3}{-6}=\frac{1}{2},\frac{c_1}{c_2}=\frac{8}{9}$

$\because \frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$,

the lines are parallel and have no solutions, i.e. the equations are an inconsistent pair

(iii) $\frac{a_1}{a_2}=\frac{\frac{3}{2}}{9}=\frac{1}{6},\frac{b_1}{b_2}=\frac{\frac{5}{3}}{-10}=-\frac{1}{6},\frac{c_1}{c_2}=\frac{7}{14}=\frac{1}{2}$

$\because \frac{a_1}{a_2}\ne \frac{b_1}{b_2}\ne \frac{c_1}{c_2}$,

the lines intersect and have an unique consistent solution.

(iv) $\frac{a_1}{a_2}=\frac{5}{-10}=-\frac{1}{2},\frac{b_1}{b_2}=\frac{-3}{6}=-\frac{1}{2},\frac{c_1}{c_2}=\frac{11}{-22}=-\frac{1}{2}$

$\because \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$,

the lines are coincident and have infinitely many solutions. The equations form a consistent pair of equations.

(v) $\frac{a_1}{a_2}=\frac{\frac{4}{3}}{2}=\frac{2}{3},\frac{b_1}{b_2}=\frac{2}{3},\frac{c_1}{c_2}=\frac{8}{12}=\frac{2}{3}$

$\because \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$,

the lines are coincident and have infinitely many solutions. The equations form a consistent pair of equations.