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Show graphically that each one of the following systems of equation has infinitely many solution:
$x\ β\ 2y\ =\ 5$
$3x\ β\ 6y\ =\ 15$
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 $y=\frac{5-5}{2}=0$
$x$ | $3$ | $5$ |
$y=\frac{x-5}{2}$ | $-1$ | $0$ |
For equation (ii),
If $x=1$ then $y=\frac{3(1)-15}{6}=\frac{-12}{6}=-2$
If $x=-1$ then $y=\frac{3(-1)-15}{6}=\frac{-3-15}{6}=\frac{-18}{6}=-3$
$x$ | $1$ | $-1$ |
$y=\frac{3x-15}{6}$ | $-2$ | $-3$ |
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.