Question

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

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.