Draw the graphs of the lines x – y = 1 and 2x + y = 8. Shade the area formed by these two lines and the y-axis.
Question:

Draw the graphs of the lines x – y = 1 and 2x + y = 8. Shade the area formed by these two lines and the y-axis. Also, find this area.

Solution:

$x-y=1$

$\Rightarrow y=x-1$

When $x=0, y=0-1=-1$

When $x=1, y=1-1=0$

When $x=2, y=2-1=1$

Thus, the points on the line x – y = 1 are as given in the following table:

Plotting the points (0, –1), (1, 0) and (21) and drawing a line passing through these points, we obtain the graph of of the line x – y = 1.

$2 x+y=8$

$\Rightarrow y=-2 x+8$

When $x=1, y=-2 \times 1+8=-2+8=6$

When $x=2, y=-2 \times 2+8=-4+8=4$

When $x=3, y=-2 \times 3+8=-6+8=2$

Thus, the points on the line  2x + y = 8 are as given in the following table:

Plotting the points (1, 6), (2, 4) and (32) and drawing a line passing through these points, we obtain the graph of of the line  2x + y = 8.

The shaded region represents the area bounded by the lines x – y = 1, 2x + y = 8 and the y-axis. This represents a triangle.

It can be seen that the lines intersect at the point C(3, 2). Draw CD perpendicular from C on the y-axis.

Height = CD = 3 units

Base = AB = 9 units

$\therefore$ Area of the shaded region $=$ Area of $\Delta \mathrm{ABC}=\frac{1}{2} \times \mathrm{AB} \times \mathrm{CD}=\frac{1}{2} \times 9 \times 3=\frac{27}{2}$ square units