Question

Show graphically that the following given systems of equations has infinitely many solutions:
$$\begin{gathered} x-2y=5 \\ 3x-6y=15 \end{gathered}$$

Answer 1

From the first equation, write y in terms of x
$y=\frac{x-5}{2}.....\left(i\right)$
Substitute different values of x in (i) to get different values of y
$$\begin{gathered} \mathrm{For}x=-5,y=\frac{-5-5}{2}=-5 \\ \mathrm{For}x=1,y=\frac{1-5}{2}=-2 \\ \mathrm{For}x=3,y=\frac{3-5}{2}=-1 \end{gathered}$$
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
$y=\frac{3x-15}{6}.....\left(ii\right)$
Now, substitute different values of x in (ii) to get different values of y
$$\begin{gathered} \mathrm{For}x=-3,y=\frac{-9-15}{6}=-4 \\ \mathrm{For}x=-1,y=\frac{-3-15}{6}=-3 \\ \mathrm{For}x=5,y=\frac{15-15}{6}=0 \end{gathered}$$
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.