Question

Show graphically that the system of equations x - 2y = 5, 3x - 6y = 15 has infinitely many solutions.

Answer 1

From the first equation, write y in terms of x 

y = `(x-5)/2`                        …….(i)
Substitute different values of x in (i) to get different values of y

For x = -5, y = `(−5−5)/2 = -5`
For x = 1, y =`( 1 − 5)/2= -2`
For x = 3, y =`( 3 − 5)/2 = -1`
Thus, the table for the first equation (x - 2y = 5) is

x	-5	1	3
y	-5	-2	-1

Now, plot the points A(-5, -5), B(1, -2) and C(3, -1) on a graph paper and join A, B and C to get the graph of x - 2y = 5.
From the second equation, write y in terms of x

y =` (3x − 15)/6`                                          …….(ii)

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

For x = -3, y = `(−9 − 15 )/6 = -4`
For x = -1, y = `(-3 − 15)/6 = -3`
For x = 5, y = (15 − 15)/6 = 0`
So, the table for the second equation (3x - 6y = 15) is

x	-3	-1	5
y	-4	-3	0

Now, plot the points D(-3, -4), E(-1, -3) and F(5, 0) on the same graph paper and join D, E and F to get the graph of 3x - 6y = 15.

From the graph, it is clear that, the given lines coincide with each other.
Hence, the solution of the given system of equations has infinitely many solutions.