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Determine graphically whether the system of equations $x\ –\ 2y\ =\ 2$, $4x\ –\ 2y\ =\ 5$ is consistent or in-consistent.
Given:
The given system of equations is:
$x\ –\ 2y\ =\ 2$
$4x-2y=5$
To do:
We have to determine whether the given system of equations is consistent or inconsistent.
Solution:
The given pair of equations are:
$x\ -\ 2y\ -\ 2\ =\ 0$....(i)
$2y=x-2$
$y=\frac{x-2}{2}$
$4x-2y-5=0$.....(ii)
$2y=4x-5$
$y=\frac{4x-5}{2}$
To represent the above equations graphically we need at least two solutions for each of the equations.
For equation (i),
If $x=2$ then $y=\frac{2-2}{2}=\frac{0}{2}=0$
If $x=0$ then $y=\frac{0-2}{2}=\frac{-2}{2}=-1$
$x$ | $0$ | $2$ |
$y=\frac{x-2}{2}$ | $-1$ | $0$ |
For equation (ii),
If $x=1$ then $y=\frac{4(1)-5}{2}=\frac{4-5}{2}=\frac{-1}{2}$
If $x=2$ then $y=\frac{4(2)-5}{2}=\frac{8-5}{2}=\frac{3}{2}$
$x$ | $1$ | $2$ |
| $y=\frac{4x-5}{2}$ | $\frac{-1}{2}$ | $\frac{3}{2}$ |
The above situation can be plotted graphically as below:
The lines AB and CD represent the equations $x–2y=2$ and $4x-2y=5$.
As we can see both the lines intersect each other.
Hence, the given system of equations is consistent.
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