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
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.
(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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