Determine by drawing graphs, whether the following system of linear equation has a unique solution or not:

$2x\ –\ 3y\ =\ 6$ and $x\ +\ y\ =\ 1$

Given:

The given system of equations is:

$2x\ –\ 3y\ =\ 6$  and  $x\ +\ y\ =\ 1$

 

To do:

We have to determine whether the given system of equations has a unique solution or not.

Solution:

The given pair of equations are:

$2x\ -\ 3y\ -\ 6\ =\ 0$....(i)

$3y=2x-6$

$y=\frac{2x-6}{3}$

$x+y-1=0$.....(ii)

$y=1-x$

To represent the above equations graphically we need at least two solutions for each of the equations.

For equation (i),

If $x=3$ then $y=\frac{2(3)-6}{3}=\frac{6-6}{3}=0$

If $x=6$ then $y=\frac{2(6)-6}{3}=\frac{12-6}{3}=2$

$x$	$3$	$6$
$y=\frac{2x-6}{3}$	$0$	$2$

For equation (ii),

If $x=1$ then $y=1-1=0$

If $x=0$ then $y=1-0=1$

$x$	$1$	$0$
$y=1-x$	$0$	$1$

The above situation can be plotted graphically as below:

 

The lines AB and CD represent the equations $2x–3y=6$ and $x+y=1$.

We can see that both the lines intersect each other at one point.

Hence, the given system of equations has a unique solution.