Show graphically that each one of the following systems of equation has infinitely many solution:

$x\ –\ 2y\ =\ 5$
$3x\ –\ 6y\ =\ 15$

Given:

The given system of equations is:

$x\ –\ 2y\ =\ 5$

$3x\ –\ 6y\ =\ 15$

To do:

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

Solution:

The given pair of equations are:

$x\ -\ 2y\ -\ 5\ =\ 0$....(i)

$2y=x-5$

$y=\frac{x-5}{2}$

$3x\ -\ 6y\ -\ 15\ =\ 0$....(ii)

$6y=3x-15$

$y=\frac{3x-15}{6}$

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

For equation (i),

If $x=3$ then $y=\frac{3-5}{2}=\frac{-2}{2}=-1$

If $x=5$ then $y=\frac{5-5}{2}=0$

For equation (ii),

If $x=1$ then $y=\frac{3(1)-15}{6}=\frac{-12}{6}=-2$

If $x=-1$ then $y=\frac{3(-1)-15}{6}=\frac{-3-15}{6}=\frac{-18}{6}=-3$

The above situation can be plotted graphically as below:

The line AB represents the equation $x-2y-5=0$ and the line PQ represents the equation $3x-6y-15=0$.

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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