Question

Solve graphically $x-3y=6$; $x-3y+9=0$

Answer 1

Let us find three points for each equation, by choosing three $x$ values and computing the corresponding $y$ values. We present our results in the tables.
Line 1: $x-3y=6$
$3y=x-6\Rightarrow y=\frac{x}{3}-2$
Substituting $x=-3,0,3$ in the above equation, we get the values of $y$ as follows

$y=\frac{x}{3}-2$

$x$	$-3$	$0$	$3$
$\frac{x}{3}$	$-1$	$0$	$1$
$y=\frac{x}{3}-2$	$-3$	$-2$	$-1$

Line 2: $x-3y+9=0$
$3y=x+9$
$y=\frac{x}{3}+3$
Substituting $x=-3,0,3$ in above equation, we get

$y=\frac{x}{3}+3$

$x$	$-3$	$0$	$3$
$\frac{x}{3}$	$-1$	$0$	$1$
$y=\frac{x}{3}+3$	$2$	$3$	$4$

We plot the points $(-3,-3)$, $(0,-2)$ and $(3,-1)$ in the graph sheet and draw the line through them. Next, we plot the points $(-3,2)$, $(0,3)$ and $(3,4)$ in the same graph sheet and draw the line through them. We find that the two graphs are parallel. That is no point is common to both lines. So, the system of equations has no solution.