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:
(i) Intersecting lines
(ii) Parallel lines
(iii) Coincident lines

Answer 1

Hint: Here, we have to find the other pair of linear equations in two variables for the equation $2x+3y-8=0$, such that they are intersecting lines, parallel lines and coincident lines. First, for the parallel lines the following condition should be satisfied, $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$, for the intersecting lines the condition to be satisfied is, $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$and for the coincident lines it should satisfy the condition, $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}$.

Complete step-by-step answer:
We can find three different equations which will pair up with the equation $2x+3y-8=0$ for the above criteria.

Here, we are given the linear equation:
 $2x+3y-8=0$ …… (1)
Now, we have another linear equation in two variables such that the geometrical representation of the pair so formed is intersecting lines, parallel lines and coincident lines.
Consider the general form for a pair of linear equations in two variables x and y:
${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ ……. (2)
${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ ……. (3)
(i)Intersecting lines:
First, we can consider the criteria for the intersecting lines. Two lines are said to be intersecting if it satisfies the given condition;
$\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ …. (4)
Here, we have one equation:
$2x+3y-8=0$
From the above equation we can say that:
 ${{a}_{1}}=2,{{b}_{1}}=3$ and ${{c}_{1}}=-8$
Now, we have to find the other line which satisfies the above condition.
Now, assume value ${{a}_{2}}$ and ${{b}_{2}}$ which satisfies the equation (3).
That is for getting another equation to meet the criterion multiply the coefficient of x with any number and multiply the coefficient of y with any other number in equation (1). So, there are many possible equations, among them one is:
$3x-2y-8$
Hence, if we apply the condition we get:
$\dfrac{2}{3}\ne \dfrac{3}{-2}$.
Therefore, we can take the other linear equation as $3x-2y-8$ which intersects the line $2x+3y-8=0$

(ii) Parallel lines
Now, we can say that two lines equation (1) and equation (2) are said to be parallel if it satisfies the given conditions:
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$
For getting any number to satisfy this criteria multiply the coefficient of x and y with the same number and multiply the coefficient of constant with any other number in equation (1).
So, we can consider the equation as $2x+3y+4=0$.
By applying the conditions we get:
$\dfrac{2}{2}=\dfrac{3}{3}\ne \dfrac{-8}{4}$
Therefore, we can say that $2x+3y+4=0$ is parallel to the line $2x+3y-8=0$.

(iii)Coincident lines
The two lines equation (1) and equation (2) are said to be coincidental lines if they satisfies the following condition:
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}$
Now, to get any equation satisfying the above criterion is multiplying the whole equation by any number.
Now, consider the equation $4x+6y-16=0$.
By applying the condition we get:
$\begin{align}
  & \dfrac{2}{4}=\dfrac{3}{6}=\dfrac{-8}{-16} \\
 & \Rightarrow \dfrac{1}{2}=\dfrac{1}{2}=\dfrac{1}{2} \\
\end{align}$
Therefore, we can say that $4x+6y-16=0$ is coincidental to the line $2x+3y-8=0$.

Note: Here, we can find the pair of linear equations which are intersecting, parallel and coincidental. If two lines intersect each other at a point then it has a unique solution. If two lines are coincident then they will have an infinite number of solutions. Similarly, if two lines are parallel to each other then, they will have no solution. Depending upon the pair of linear equations we can say that the solution is consistent or inconsistent. A pair of linear equations which has a unique solution or infinite number of solutions is said to be consistent. A pair of linear equations which has no solution is said to be inconsistent.