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Solve the following system of linear equations graphically and shade the region between the two lines and x-axis:
$3x\ +\ 2y\ -\ 4\ =\ 0$
$2x\ -\ 3y\ -\ 7\ =\ 0$
Given:
The given system of equations is:
$3x\ +\ 2y\ -\ 4\ =\ 0$
$2x\ -\ 3y\ -\ 7\ =\ 0$
To do:
We have to solve the given system of equations and shade the region between the two lines and x-axis.
Solution:
The given pair of equations is:
$3x\ +\ 2y\ -\ 4\ =\ 0$....(i)
$2y=4-3x$
$y=\frac{4-3x}{2}$
$2x-3y-7=0$.....(ii)
$3y=2x-7$
$y=\frac{2x-7}{3}$
To represent the above equations graphically we need at least two solutions for each of the equations.
For equation (i),
If $x=2$ then $y=\frac{4-3(2)}{2}=\frac{4-6}{2}=\frac{-2}{2}=-1$
If $x=0$ then $y=\frac{4-3(0)}{2}=\frac{4}{2}=2$
$x$ | $2$ | $0$ |
$y=\frac{4-3x}{2}$ | $-1$ | $2$ |
For equation (ii),
If $x=2$ then $y=\frac{2(2)-7}{3}=\frac{4-7}{3}=\frac{-3}{3}=-1$
If $x=-1$ then $y=\frac{2(-1)-7}{3}=\frac{-2-7}{3}=\frac{-9}{3}=-3$
$x$ | $2$ | $-1$ |
| $y=\frac{2x-7}{3}$ | $-1$ | $-3$ |
The above situation can be plotted graphically as below:
The lines AB and CD represent the equations $3x+2y-4=0$ and $2x-3y-7=0$.
The solution of the given system of equations is the intersection point of the lines AB and CD.
Hence, the solution of the given system of equations is $x=2$ and $y=-1$.
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