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

$2y\ =\ 4x\ –\ 6$ and $2x\ =\ y\ +\ 3$

Given:

The given system of equations is:

$2y\ =\ 4x\ –\ 6$  and  $2x\ =\ y\ +\ 3$

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:

$4x\ -\ 2y\ -\ 6\ =\ 0$....(i)

$2y=4x-6$

$y=\frac{4x-6}{2}$

$2x-y-3=0$.....(ii)

$y=2x-3$

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

For equation (i),

If $x=1$ then $y=\frac{4(1)-6}{2}=\frac{4-6}{2}=-1$

If $x=2$ then $y=\frac{4(2)-6}{2}=\frac{8-6}{2}=1$

For equation (ii),

If $x=1$ then $y=2(1)-3=2-3=-1$

If $x=2$ then $y=2(2)-3=4-3=1$

The above situation can be plotted graphically as below:

The lines AB and CD represent the equations $2y=4x-6$ and $2x=y+3$.

We can see that both equations represent the same line.

Hence, the given system of equations does not have a unique solution.

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Updated on: 10-Oct-2022

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