Question

On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point and parallel or coincide:
$9x+3y+12=0$ $18x+6y+24=0$

Answer 1

$9x+3y+12=0⟶\left(i\right)18x+6y+24=0⟶\left(ii\right)$

$9x+3y+12=0$

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

we get ,

$a_1=9,b_1=3,c_1=12$

$18x+6y+24=0$

Comparing with $a_{2}x+b_{2}y+c_2=0$

we get ,

$a_2=18,b_2=6$ and $c_2=24$

$\therefore \frac{a_1}{a_2}=\frac{9}{18}=\frac{1}{2}\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}\frac{c_1}{c_2}=\frac{12}{24}=\frac{1}{2}$

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

We have infinite solutions .Therefore , the lines that represent linear equations are coincident.