Show graphically that each one of the following systems of equation has infinitely many (i)2x + 3y = 6(ii)4x + 6y = 12
Show graphically that each one of the following systems of equation has infinitely many (i)2x + 3y = 6(ii)4x + 6y = 12

Given,

2x + 3y = 6……. (i)

4x + 6y = 12……. (ii)

For equation (i),

⇒ y = (6 - 2x) /3

When x = 0, we have y = (6 - 2(0))/3 = 2

When x = 3, we have y = (6 - 2(3))/3 = 0

Thus we have the following table giving points on the line 2x + 3y = 6

x03
y20

For equation (ii),

We solve for y:

⇒ y = (12 - 4x)/6

So, when x = 0

y = (12 - 4(0))/6 = 2

And, when x = 3

⇒ y = (12 - 4(3))/6 = 0

Thus we have the following table giving points on the line 4x + 6y = 12

x03
y20

Graph of the equations (i) and (ii) is as below:

Thus, the graphs of the two equations are coincident.

Hence, the system of equations has infinitely many solutions.