Question

Shade the region formed by the graph of 2x – y = 4, x + y = 2 and the y-axis. Write the coordinates of the vertices of the figure obtained.

Answer 1

Hint: To solve this question, we will first draw the graph of $2x – y = 4$ and similarly separate draw the graph of $x + y = 2$ and then join these two with the y-axis. We will form a figure and then try to determine the coordinates of the points obtained.

Complete step-by-step solution:
Let us first consider, $2x – y = 4$, and draw its graph.
When $x = 0$, we get,
\[2\times 0-y=4\]
\[\Rightarrow y=-4\]
Therefore, $x = 0, y = – 4.$
And when we put $y = 0,$ we get,
\[2x=4\]
\[\Rightarrow x=\dfrac{4}{2}=2\]
Therefore, $x = 2, y = 0.$
So, we have, $(0, – 4)$ and $(2, 0)$ as the coordinates. So, we will draw the line $2x – y = 4$ using the coordinates $(0, – 4)$ and $(2, 0)$.

Similarly, we will draw the line $x + y = 2$. When $x = 0, y = 2.$ Therefore, the coordinates we get is (0, 2) is a point on the line $x + y = 2$. And when $y = 0, x = 2$, so $(2, 0)$ is also a point on the line $x + y = 2. $
Let us draw the line $x + y = 2$ using the coordinates $(0, 2)$ and $(2, 0).$ We have,

Now, let us mix the graphs of $2x – y = 4$ and $x + y = 2$ and the y-axis, all together to form a triangle. Doing so, we will get,

The shaded region is the required figure obtained. Let the vertices be A, B, and C of the triangle formed. Then as observed by the figure, $A = (0, 2), B = (2, 0),$ and $C = (0, – 4)$ and thus the vertices of the figure are obtained.

Note: If in this question, we were given to take the x-axis apart from the y-axis, then we will need one more information that is exactly which part of the x-axis. The part having a positive y-axis or the part having a negative y-axis. So, always remember to check which axis is given.