Question

Show graphically that the following given systems of equations has infinitely many solutions:
$$\begin{gathered} 2x+3y=6 \\ 4x+6y=12 \end{gathered}$$

Answer 1

From the first equation, write y in terms of x
$y=\frac{6-2x}{3}.....\left(i\right)$
Substitute different values of x in (i) to get different values of y
$$\begin{gathered} \mathrm{For}x=-3,y=\frac{6+6}{3}=4 \\ \mathrm{For}x=3,y=\frac{6-6}{3}=0 \\ \mathrm{For}x=6,y=\frac{6-12}{3}=-2 \end{gathered}$$
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
$y=\frac{12-4x}{6}.....\left(ii\right)$
Now, substitute different values of x in (ii) to get different values of y
$$\begin{gathered} \mathrm{For}x=-6,y=\frac{12+24}{6}=6 \\ \mathrm{For}x=0,y=\frac{12-0}{6}=2 \\ \mathrm{For}x=9,y=\frac{12-36}{6}=-4 \end{gathered}$$
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 coincide with each other.
Hence, the solution of the given system of equations has infinitely many solutions.