Question

 Show graphically that the system of equations 2x + 3y = 4, 4x + 6y = 12 is inconsistent.

Show graphically that the following given system of equations is inconsistent, i.e., has no solution:

2x + 3y = 4, 4x + 6y = 12

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 + 3y = 4

2x + 3y = 4

⇒ 3y = (–2x + 4)   ...(i)

Putting x = 2, we get y = 0

Putting x = –1, we get y = 2

Putting x = –4, we get y = 4

Thus, we have the following table for the equation 2x + 3y = 4.

x	2	–1	–4
y	0	2	4

Now, plot the points A(2, 0), B(–1, 2) and C(–4, 4) on the graph paper.

Join AB and BC to get the graph line AC. Extend it on both ways.

Thus, the line AC is the graph of 2x + 3y = 4.

Graph of 4x + 6y = 12

4x + 6y = 12

⇒ 6y = (–4x + 12)

⇒ `y = (-4x + 12)/6`   ...(ii)

Putting x = 3, we get y = 0

Putting x = 0, we get y = 2

Putting x = 6, we get y = –2

Thus, we have the following table for the equation 4x + 6y = 12.

x	3	0	6
y	0	2	–2

Now, on the same graph, plot the points A(3, 0), B(0, 2) and C(6, –2).

Join PQ and PR to get the graph line QR. Extend it on both ways.

Thus, QR is the graph of the equation 4x + 6y = 12.

It is clear from the graph that these two lines are parallel and do not intersect when produced.

Hence, the given system of equations is inconsistent.