Question

Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not :

2 y = 4 x - 6 2 x = y + 3

Accepted Answer

The equations of graphs is

$2 y = 4 x - 6$

$4 x - 2 y = 6 \dots . . (i)$

$2 x = y + 3$

$2 x - y = 3 \dots . . (i i)$

Putting $x = 0$ in equation $(i)$ , we get

$\Rightarrow 4 \times 0 - 2 y = 6$

$\Rightarrow y = - 3$

$\Rightarrow x = 0, y = - 3$

Putting $y = 0$ in equation $(i)$ , we get

$\Rightarrow 4 x - 2 \times 0 = 6$

$\Rightarrow x = 3 / 2$

$x = 3 / 2, y = 0$

Use the following table to draw the graph:

$x$ $0$ $\frac{3}{2}$

$y$ $- 3$ $0$

The graph of $(i)$ can be obtained by plotting the two points $A (0, - 3), B (\frac{3}{2},0)$

Graph of the equation $(i i)$

$2 x - y = 3 . . . . . (i i)$

Putting $x = 0$ in equation $(i i)$ , we get

$\Rightarrow 2 \times 0 - y = 3$

$\Rightarrow y = - 3$

$x = 0, y = - 3$

Putting $y = 0$ in equation $(i i)$ , we get

$\Rightarrow 2 x - 0 = 3$

$\Rightarrow x = \frac{3}{2}$

$x = \frac{3}{2}, y = 0$

Use the following table to draw the graph

$x$ $0$ $\frac{3}{2}$

$y$ $- 3$ $0$

Draw the graph by plotting the two points $C (0, - 3), D (\frac{3}{2},0)$ from table.

From the graph, we observe that the two lines are coincident.

Consequently, every solution of one equation is a solution of the other.

Hence, the system of equations has infinitely many solutions.

R. D. Sharma Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 2: EXERCISE 3.2

R. D. Sharma Mathematics Solutions for Exercise - R. D. Sharma Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 2: EXERCISE 3.2

Questions from R. D. Sharma Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 2: EXERCISE 3.2 with Hints & Solutions

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