Given the linear equation 2x + 3y - 8 = 0, write another linear e…

Question

Given the linear equation $2x + 3y - 8 = 0,$ write another linear equation in two variables such that the geometrical representation of the pair so formed is:

Intersecting lines.
Parallel lines.
Coincident lines.

✓

Answer

For the two lines $a_1x + b_1x + c_1 = 0$ and $a_2 x+b_2 x+c_2=0$, to be intersecting, we must have
$\frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}$
So, the other linear equation can be $5x + 6y - 16 = 0$
as $\frac{\text{a}_1}{\text{a}_2}=\frac{2}{5},$
$\frac{\text{b}_1}{\text{b}_2}=\frac{3}{6}=\frac{1}{2},$
$\frac{\text{c}_1}{\text{c}_2}=\frac{-8}{-16}=\frac{1}{2}$
For the two lines $a_1x + b_1x + c_1= 0$ and $a_2 x+b_2 x+c_2=0$, to be parallel, we must have
$\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
So, the other linear equation can be $6x + 9y + 24 =0$
as $\frac{\text{a}_1}{\text{a}_2}=\frac{2}{6}=\frac{1}{3},$
$\frac{\text{b}_1}{\text{b}_2}=\frac{3}{9}=\frac{1}{3},$
$\frac{\text{c}_1}{\text{c}_2}=\frac{-8}{24}=\frac{-1}{3}$
For the two lines $a_1x + b_1x+ c_1= 0$ and $a_2 x+b_2 x+c_2=0$, to be coincident, we must have
$\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
So, the other linear equation can be $8x + 12y - 32 =0$
as $\frac{\text{a}_1}{\text{a}_2}=\frac{2}{8}=\frac{1}{4},$
$\frac{\text{b}_1}{\text{b}_2}=\frac{3}{12}=\frac{1}{4},$
$\frac{\text{c}_1}{\text{c}_2}=\frac{-8}{-32}=\frac{1}{4}$

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