Show graphically that each one of the following systems of equation has infinitely many solution:
$3x\ +\ y\ =\ 8$
$6x\ +\ 2y\ =\ 16$

Given:

The given system of equations is:

$3x\ +\ y\ =\ 8$

$6x\ +\ 2y\ =\ 16$

To do:

We have to show that the above system of equations has infinitely many solutions.

Solution:

The given pair of equations are:

$3x\ +\ y\ -\ 8\ =\ 0$....(i)

$y=8-3x$

$6x\ +\ 2y\ -\ 16\ =\ 0$....(ii)

$2y=16-6x$

$y=\frac{16-6x}{2}$

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

For equation (i),

If $x=2$ then $y=8-3(2)=8-6=2$

If $x=3$ then $y=8-3(3)=8-9=-1$

For equation (ii),

If $x=3$ then $y=\frac{16-6(3)}{2}=\frac{-2}{2}=-1$

If $x=2$ then $y=\frac{16-6(2)}{2}=\frac{16-12}{2}=\frac{4}{2}=2$

The above situation can be plotted graphically as below:

The lines AB and PQ represent the equations $3x+y-8=0$ and $6x+2y-16=0$.

As we can see, both equations represent the same line.

Hence, the given system of equations has infinitely many solutions.

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Updated on: 10-Oct-2022

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