Question

Show graphically that the following given systems of equations is inconsistent i.e. has no solution:
$$\begin{gathered} 2x+y=6 \\ 6x+3y=20 \end{gathered}$$

Answer 1

From the first equation, write y in terms of x
$y=6-2x.....\left(i\right)$
Substitute different values of x in (i) to get different values of y
$$\begin{gathered} \mathrm{For}x=0,y=6-0=6 \\ \mathrm{For}x=2,y=6-4=2 \\ \mathrm{For}x=4,y=6-8=-2 \end{gathered}$$
Thus, the table for the first equation (2x + y = 6) is

x	0	2	4
y	6	2	−2

Now, plot the points A(0,6), B(2,2) and C(4,−2) on a graph paper and join
A, B and C to get the graph of 2x + y = 6.
From the second equation, write y in terms of x
$y=\frac{20-6x}{3}.....\left(ii\right)$
Now, substitute different values of x in (ii) to get different values of y
$$\begin{gathered} \mathrm{For}x=0,y=\frac{20-0}{3}=\frac{20}{3} \\ \mathrm{For}x=\frac{10}{3},y=\frac{20-20}{3}=0 \\ \mathrm{For}x=5,y=\frac{20-30}{3}=-\frac{10}{3} \end{gathered}$$
So, the table for the second equation (6x + 3y = 20 ) is

x	0	$\frac{10}{3}$	5
y	$\frac{20}{3}$	0	$-\frac{10}{3}$

Now, plot the points $D\left(0,\frac{20}{3}\right),E\left(\frac{10}{3},0\right)\mathrm{and}F\left(5,-\frac{10}{3}\right)$ on the same graph paper and join
D, E and F to get the graph of 6x + 3y = 20.



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.