Question

Show graphically that the system of equations 2x + 3y = 6, 4x + 6y = 12 has infinitely many solutions. 

Show graphically that the following given system of equations has infinitely many solutions:

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

Answer 1

From the first equation, write y in terms of x

`y = (6 - 2x)/3`   ...(i)

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

For x = –3, y = `(6 + 6)/3` = 4

For x = 3, y = `(6 - 6)/3` = 0

For x = 6, y = `(6 - 12)/3` = –2

Thus, the table for the first equation (2x + 3y = 6) is

x	–3	3	6
y	4	0	–2

Now, plot the points A(–3, 4), B(3, 0) and C(6, –2) on a graph paper and join A, B and C to get the graph of 2x + 3y = 6.

From the second equation, write y in terms of x

`y = (12 - 4x)/6`   ...(ii)

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

For x = –6, y = `(12 + 24)/6` = 6

For x = 0, y = `(12 - 0)/6` = 2

For x = 9, y = `(12 - 36)/6` = –4

So, the table for the second equation (4x + 6y = 12) is

x	–6	0	9
y	6	2	–4

Now, plot the points D(–6, 6), E(0, 2) and F(9, –4) on the same graph paper and join D, E and F to get the graph of 4x + 6y = 12.

From the graph, it is clear that, the given lines coincidence with each other.

Hence, the solution of the given system of equations has infinitely many solutions.