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:$\left( i \right)$ Intersecting lines$\left( {ii} \right)$ Parallel lines$\left( {iii} \right)$ Coincident lines

Hint: - Directly use the criteria of slopes for intersecting, parallel and coincident lines in coordinate geometry to solve the question.

$\left( i \right)$ Intersecting lines
For intersecting line, the linear equations should meet following condition:
$\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
For getting another equation to meet this criterion, multiply the coefficient of $x$ with any number and multiply the coefficient of $y$ with any other number.
In order to get two intersecting lines or to achieve the above criteria let us multiply the coefficient of $x$ with $2$ and multiply the coefficient of $y$ with $3$.
A possible equation can be as follows:
$4x + 9y - 8 = 0$

$\left( {ii} \right)$ Parallel lines
For parallel lines, the linear equation should meet following condition:
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$
For getting another equation to meet this criterion, multiply the coefficient of $x$ and $y$ with the same number and multiply the constant term with any other number.
In order to get two parallel lines or to achieve the above criteria let us multiply the coefficient of $x$ and $y$ with $2$ and multiply the constant term with $3$.
A possible equation can be as follows:
$4x + 6y - 24 = 0$

$\left( {iii} \right)$ Coincident lines
For coincident lines, the linear equation should meet following condition:
$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$
For getting another equation to meet this criterion, multiply the whole equation with any number.
In order to get two coincident lines or to achieve the above criteria let us multiply the whole equation by $2$.
A possible equation can be as follows:
$4x + 6y - 16 = 0$

Note: If two linear equations have the same slope, then lines will be parallel and they will have no solution. If two linear equations representing lines are coincident, then lines are the same and will have infinitely many solutions. In case of intersecting lines, there will be only one solution.