Question

Draw the graphs of the lines 2x + y = 6 and 2x – y + 2 = 0. Shade the region bounded by these lines and the x-axis. Find the area of the shaded region.

Answer 1

$$\begin{gathered} 2x+y=6 \\ \Rightarrow y=-2x+6 \end{gathered}$$
When x = 0, $y=-2\times 0+6=0+6=6$
When x = 1, $y=-2\times 1+6=-2+6=4$
When x = 2, $y=-2\times 2+6=-4+6=2$

Thus, the points on the line 2x + y = 6 are as given in the following table:

x	0	1	2
y	6	4	2

Plotting the points (0, 6), (1, 4) and (2, 2) and drawing a line passing through these points, we obtain the graph of of the line 2x + y = 6.

$$\begin{gathered} 2x-y+2=0 \\ \Rightarrow y=2x+2 \end{gathered}$$
When x = 0, $y=2\times 0+2=0+2=2$
When x = 1, $y=2\times 1+2=2+2=4$
When x = –1, $y=2\times \left(-1\right)+2=-2+2=0$

Thus, the points on the line 2x – y + 2 = 0 are as given in the following table:

x	0	1	–1
y	2	4	0

Plotting the points (0, 2), (1, 4) and (–1, 0) and drawing a line passing through these points, we obtain the graph of of the line 2x – y + 2 = 0.



The shaded region represents the area bounded by the lines 2x + y = 6, 2x – y + 2 = 0 and the x-axis. This represents a triangle.

It can be seen that the lines intersect at the point C(1, 4). Draw CD perpendicular from C on the x-axis.

Height = CD = 4 units

Base = AB = 4 units

∴ Area of the shaded region = Area of ∆ABC = $\frac{1}{2}\times \mathrm{AB}\times \mathrm{CD}=\frac{1}{2}\times 4\times 4$ = 8 square units