2017 AMC 10B Problem 10

The first equation can be rewritten as $y=\dfrac a2x-\dfrac c2,$ and the second as $y=-\dfrac2b x-\dfrac cb.$ Because the lines are perpendicular, their slopes multiply to $-1,$ so $a=b.$ Substituting $(x,y)=(1,-5)$ into the two original equations gives $a+10=c$ and $2-5a=-c.$ Adding these equations yields $12-4a=0,$ so $a=3$ and $c=13.$ Thus, the correct answer is E .