Question

Show graphically that the system of equations 2x + y = 6, 6x + 3y = 20 is inconsistent.

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

2x + y = 6, 6x + 3y = 20

Answer 1

From the first equation, write y in terms of x

y = 6 – 2x   ...(i)

Substitute different values of x in (i) to get different values of y

For x = 0, y = 6 – 0 = 6

For x = 2, y = 6 – 4 = 2

For x = 4, y = 6 – 8 = –2

Thus, the table for the first equation (2x + y = 6) is

x	0	2	4
y	6	2	–2

Now, plot the points A(0, 6), B(2, 2) and C(4, –2) on a graph paper and join A, B and C to get the graph of 2x + y = 6.

From the second equation, write y in terms of x

`y = (20 - 6x)/3`   ...(ii)

Now, substitute different values of x in (ii) to get different values of y

For x = 0, y = `(20 - 0)/3 = 20/3`

For x = `10/3`, y = `(20 - 20)/3` = 0

For x = 5, y = `(20 - 30)/3 = -10/3`

So, the table for the second equation (6x + 3y = 20) is

x	0	`10/3`	5
y	`20/3`	0	`-10/3`

Now, plot the points D`(0, 20/3)`, O`(10/3, 0)` and E`(5, -10/3)` on the same graph paper and join D, E and F to get the graph of 6x + 3y = 20.

From the graph, it is clear that, the given lines do not intersect at all when produced.

Hence, the system of equations has no solution and therefore is inconsistent.