Show graphically that each one of the following systems of equations is inconsistent (i.e. has no solution):

$3x\ –\ 5y\ =\ 20$
$6x\ –\ 10y\ =\ – 40$

Given:

The given system of equations is:

$3x\ –\ 5y\ =\ 20$

$6x\ –\ 10y\ =\ – 40$

To do:

We have to show that the above system of equations is inconsistent.

Solution:

The given pair of equations are:

$3x\ -\ 5y\ -\ 20\ =\ 0$....(i)

$5y=3x-20$

$y=\frac{3x-20}{5}$

$6x\ -\ 10y\ +\ 40\ =\ 0$....(ii)

$10y=6x+40$

$y=\frac{6x+40}{10}$

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

For equation (i),

If $x=5$ then $y=\frac{3(5)-20}{5}=\frac{15-20}{5}=\frac{-5}{5}=-1$

If $x=0$ then $y=\frac{3(0)-20}{5}=\frac{-20}{5}=-4$

For equation (ii),

If $x=0$ then $y=\frac{6(0)+40}{10}=\frac{40}{10}=4$

If $x=-5$ then $y=\frac{6(-5)+40}{10}=\frac{-30+40}{10}=\frac{10}{10}=1$

The above situation can be plotted graphically as below:

The lines AB and PQ represent the equations $3x-5y-20=0$ and $6x-10y+40=0$.

As we can see, there is no common point between the two lines.

Hence, the given system of equations is inconsistent.

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Updated on: 10-Oct-2022

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