Coordinates

Writing each equation in slope-intercept form, we get $y=\frac{a}{2}x-\frac{1}{2}c$ and $y=-\frac{2}{b}x-\frac{c}{b}$ . We observe the slope of each equation is $\frac{a}{2}$ and $-\frac{2}{b}$ , respectively. Because the slope of a line perpendicular to a line with slope $m$ is $-\frac{1}{m}$ , we see that $\frac{a}{2}=-\frac{1}{-\frac{2}{b}}$ because it is given that the two lines are perpendicular. This equation simplifies to $a=b$ .

Because $(1, -5)$ is a solution of both equations, replace $x$ with $1$ and $y$ with $-5$ , and replace $b$ with $a$ , giving us $a+10=c$ and $2-5a=-c$ . Solving this system of equations, we will get $c =13$ .