Question

Show graphically that the following given systems of equations is inconsistent i.e. has no solution:
$$\begin{gathered} x-2y=6 \\ 3x-6y=0 \end{gathered}$$

Answer 1

From the first equation, write y in terms of x
$y=\frac{x-6}{2}.....\left(i\right)$
Substitute different values of x in (i) to get different values of y
$$\begin{gathered} \mathrm{For}x=-2,y=\frac{-2-6}{2}=-4 \\ \mathrm{For}x=0,y=\frac{0-6}{2}=-3 \\ \mathrm{For}x=2,y=\frac{2-6}{2}=-2 \end{gathered}$$
Thus, the table for the first equation (x − 2y = 6) is

x	−2	0	2
y	−4	−3	−2

Now, plot the points A(−2,−4), B(0,−3) and C(2,−2) on a graph paper and join
A, B and C to get the graph of x − 2y = 6.
From the second equation, write y in terms of x
$y=\frac{1}{2}x.....\left(ii\right)$
Now, substitute different values of x in (ii) to get different values of y
$$\begin{gathered} \mathrm{For}x=-4,y=\frac{-4}{2}=-2 \\ \mathrm{For}x=0,y=\frac{0}{2}=0 \\ \mathrm{For}x=4,y=\frac{4}{2}=2 \end{gathered}$$
So, the table for the second equation (3x − 6y = 0 ) is

x	−4	0	4
y	−2	0	2

Now, plot the points D(−4,−2), O(0,0) and E(4,2) on the same graph paper and join
D, E and F to get the graph of 3x − 6y = 0.



From the graph it is clear that, the given lines do not intersect at all when produced.
Hence, the system of equations has no solution and therefore is inconsistent.