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

$2x\ +\ 3y\ =\ 6$
$4x\ +\ 6y\ =\ 12$

Given:

The given system of equations is:

$2x\ +\ 3y\ -\ 6\ =\ 0$

$4x\ +\ 6y\ –\ 12\ =\ 0$

To do:

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

Solution:

The given pair of equations are:

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

$3y=6-2x$

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

$4x\ +\ 6y\ -\ 12\ =\ 0$....(ii)

$6y=12-4x$

$y=\frac{12-4x}{6}$

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

For equation (i),

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

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

For equation (ii),

If $x=0$ then $y=\frac{12-4(0)}{6}=\frac{12}{6}=2$

If $x=3$ then $y=\frac{12-4(3)}{6}=\frac{12-12}{6}=\frac{0}{6}=0$

The above situation can be plotted graphically as below:

The line AB represents the equation $2x+3y-6=0$ and the line PQ represents the equation $4x+6y-12=0$.

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

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

Tutorialspoint

Updated on: 10-Oct-2022

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