Given the linear equation 2x + 3y – 8 = 0
Question.

Given the linear equation $2 x+3 y-8=0$, 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


Solution:

(i) 2x + 3y – 8 = 0 (Given equation)

3x + 2y + 4 = 0 (New equation)

Here, $\frac{\mathbf{a}_{1}}{\mathbf{a}_{2}} \neq \frac{\mathbf{b}_{1}}{\mathbf{b}_{2}}$

Hence, the graph of the two equations will be two intersecting lines

(ii) 2x + 3y – 8 = 0 (given equation)

4x + 6y – 10 = 0 (New equation)

Here, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$

Hence, the graph of the two equations will be two parallel lines.

(iii) 2x + 3y – 8 = 0 (given equation)

4x + 6y – 16 = 0 (New equation)

Here, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

Hence, the graph of the two equations will be two conicident lines.
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