15Example: Elimination by equation coefficients

Apply the elimination

The elimination by equating coefficients ensures that only one of the two variables remains.

To do this, you have just solved both equations for $y$. You can then set equal both equations, since in a linear system of equations the variables $x$ and $y$ satisfy both equations $\mathrm{I}$ and $\mathrm{II}$.

$\def\arraystretch{1.25} \begin{array}{rrll}\mathrm{I} &=& \mathrm{II}\\5x -1& =& -\frac{1}{5}x -1 &|+1\\5x &=& -\frac{1}{5}x &|+\frac{1}{5}x\\5x + \frac{1}{5}x &=& 0 \\5 \frac{1}{5}x &=& 0 \\x &=& 0\end{array}$

Now plug $x$ into $\mathrm{I}$ or $\mathrm{II}$. To make the calculation easier, we may take the first equation.

$\mathrm{I} \qquad y = 5 \cdot \color{#CC0000}{0} -1$

$\mathrm{I}\qquad y=-1$

So the intersection point of both lines is $P \; (0|-1)$.