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 of intersecting lines:
(A) $2x + 3y + 9 = 0$
(B) $6x + 9y + 9 = 0$
(C) $3x + 2y + 9 = 0$
(D) None of these

Answer 1

Hint: For two linear equations ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ to represent a pair of intersecting lines, the condition $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ must hold. Therefore we can get infinitely many lines intersecting the given line, $2x + 3y - 8 = 0$. Consider the equation of lines given in options. The lines that are satisfying the above condition will be our answer.

Complete step by step answer:
We know that, for two linear equations ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ to represent a pair of intersecting lines:
$ \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$
The linear equation given in the question is $2x + 3y - 8 = 0$. It represents a straight line.
If we consider option (A), then the other line is $2x + 3y + 9 = 0$.
In this case:
$
   \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{2} = 1,\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{3}{3} = 1, \\
   \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \\
$
Thus, this option is incorrect.
If we consider option (B), then the other line is $6x + 9y + 9 = 0$.
In this case:
$
   \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{6} = \dfrac{1}{3},\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{3}{9} = \dfrac{1}{3}, \\
   \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \\
$
Thus, this option is also incorrect.
If we consider option (C), then the other line is $3x + 2y + 9 = 0$.
In this case:
$
   \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{3},\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{3}{2}, \\
   \Rightarrow \dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}} \\
$

Thus, $3x + 2y + 9 = 0$ is a linear equation such that the geometrical representation of the pair so formed is of intersecting lines.

Hence, (C) is the correct option.

Note: If two lines ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ are such that $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$, then the equations will represent a same straight line with infinitely many solutions.
But if they are such that $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$, then the equations will represent parallel lines with no solution.