Question

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

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 =${\displaystyle \frac{-4x+12}{6}}$                      …(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.