Question

Solve the following system of linear equations graphically:
$2x-y=1,x-y=-1.$
Shade the region bounded by these lines and the y-axis.

Answer 1

On a graph paper, draw a horizontal line X'OX and a vertical line YOY' as the x-axis and y-axis, respectively.
Graph of 2x − y = 1
2x − y = 1
⇒ y = (2x − 1) ........(i)
Putting x = 1, we get y = 1.
Putting x = 2, we get y = 3.
Putting x = 0, we get y = −1.
Thus, we have the following table for the equation 2x − y = 1.

x	1	2	0
y	1	3	−1

Now, plot the points A(1, 1), B( 2, 3) and C(0, −1) on the graph paper.
Join AB and AC to get the graph line BC. Extend it on both ways.
Thus, BC is the graph of 2x − y = 1.

Graph of x − y = −1
x − y = −1
⇒ y = (x + 1) ............(ii)
Putting x = 1, we get y = 2.
Putting x = 2, we get y = 3.
Putting x = 0, we get y = 1.
Thus, we have the following table for the equation x − y = −1.

x	1	2	0
y	2	3	1

Now, plot the points P(1, 2) and Q(0, 1). The point B(2, 3) has already been plotted. Join PB and PQ to get the graph line BQ. Extend it on both ways.
Then, BQ is the graph of the equation x − y = −1.

The two graph lines intersect at B(2, 3).
∴ The solution of the given system of equations is x = 2 and y = 3.
These graph lines intersect the y-axis at C and Q.
Hence, the region bounded by these lines and the y-axis has been shaded.