Question

Solve the following system of equations graphically:
$4x-y=4,3x+2y=14.$
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 4x − y = 4

4x − y = 4
⇒ y = (4x − 4) ........(i)
Putting x = 0, we get y = −4.
Putting x = 1, we get y = 0.
Putting x = 2, we get y = 4.
Thus, we have the following table for the equation 4x − y = 4.

x	0	1	2
y	−4	0	4

Now, plot the points A(0, −4), B( 1, 0) and C(2, 4) on the graph paper.
Join AB and BC to get the graph line AC. Extend it on both ways.
Thus, AC is the graph of 4x − y = 4.

Graph of 3x + 2y = 14
3x + 2y = 14
⇒ 2y = (14 − 3x)
∴ $y=\frac{14-3x}{2}$ ...........(ii)
Putting x = 0, we get y = 7.
Putting x = 2, we get y = 4.
Putting x = 4 we get y = 1.
Thus, we have the following table for the equation 3x + 2y = 14.

x	0	2	4
y	7	4	1

Now, plots the points P(0, 7) and Q(4, 1). The point C(2, 4) has already been plotted. Join PC and CQ to get the graph line PQ. Extend it on both ways.
Then, PQ is the graph of the equation 3x + 2y = 14.

The two graph lines intersect at C(2, 4).
∴ The solution of the given system of equations is x = 2 and y = 4.

These graph lines intersect the y-axis at P and A.
Hence, the region bounded by these lines and the y-axis has been shaded.