Question

Show graphically that the system of equations $2y-x=9,4y-2x=20$ 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 2y − x = 9
2y − x = 9
⇒ 2y = (x + 9)
∴ $y=\frac{x+9}{2}$.........(i)
Putting x = 1, we get y = 5.
Putting x = −1, we get y = 4.
Putting x = −3, we get y = 3.
Thus, we have the following table for the equation 2y − x = 9.

x	1	−1	−3
y	5	4	3

Now, plot the points A(1, 5), B( −1, 4) and C(−3, 3) 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 2y − x = 9.

Graph of 4y − 2x = 20
4y − 2x = 20
⇒ 4y = (2x + 20)
∴ $y=\frac{2x+20}{4}$...........(ii)
Putting x = 0, we get y = 5.
Putting x = 2, we get y = 6.
Putting x = −2, we get y = 4.
Thus, we have the following table for the equation 4y − 2x = 20.

x	0	2	−2
y	5	6	4

Now, on the same graph, plots the points P(0, 5), Q(2, 6) and R(−2, 4).
Join PQ and PR to get the graph line QR. Extend it on both ways.
Then, QR is the graph of the equation 4y − 2x = 20.

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.