Lines Intersection Sum
Let $ P=(a,b) $ be the point of intersection of the line $ y=2x-10 $ and the line through $ (7,8) $ and $ (9,0) $. Compute $ a+b $.
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Solution
The slope of the line through $ (7,8) $ and $ (9,0) $ is $ \frac{8-0}{7-9}=\frac{8}{-2}=-4 $. Thus, the line has equation $ y=-4x+b $ for some $ b $. Since $ B(9,0) $ lies on this line, we have $ 0=-4(9)+b \Rightarrow b=36 $, and thus the equation of the line is $ y=-4x+36 $.
To determine the point of intersection between the lines having equations $ y=-4x+36 $ and $ y=2x-10 $, we set the two expressions for $ y $ equal to each other and solve for $ x $. We have $ -4x+36=2x-10 \Rightarrow x = \frac{23}{3} $. It follows that $ y=2x-10=2\left(\frac{23}{3}\right)-10 = \frac{46}{3}-\frac{30}{3}=\frac{16}{3} $.
Thus, $ P=(\frac{23}{3},\frac{16}{3}) $ and $ a+b=\frac{23}{3}+\frac{16}{3}=\frac{39}{3}=\boxed{13} $.
Freeflows