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

Answer 1

$5x-3y-11=0⟶\left(1\right)$

$-10x+6y+22=0⟶\left(2\right)$

$5x-3y-11=0$

Comparing with $a_{1}x+b_{1}y+c_1=0$

$a_1=5,b_1=-3,c_1=-11$

$-10x+6y+22=0$

comparing with $a_{2}x+b_{2}y+c_2=8$

$a_2=-10,b_2=6,c_2=22$

$a_1=5,b_1=-3,c_1=-11$

$a_2=-10,b_2=6,c_2=22$

$\frac{a_1}{a_2}=\frac{-1}{2}$ $\frac{b_1}{b_2}=\frac{-1}{2}$ $\frac{c_1}{c_2}=\frac{-1}{2}$

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

They have infinitely many solutions.

Therefore, system of equation is constant and can be solved graphically.