Line Shift Equation
A line $ l $ is the graph of $ y = ax + b $. Line $ l_1 $ is $ l $ shifted 3 units to the left and two units downward and passes through the points $ (5,8) $ and $ (7,14) $. What is $ a - b $?
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Solution
We find the equation of $ l_1 $ first. The slope of $ l_1 $ is \[\frac{14 - 8}{7-5} = 3.\]Since $ l_1 $ passes through $ 5,8 $, we can set $ 8 = 3(5) + t $ where $ (0,t) $ is the $ y $-intercept. Subtracting 15 from both sides, we find the $ t = -7 $, and so the $ y $-intercept is $ (0,-7) $. Thus, $ l_1 $ has equation $ y = 3x - 7 $.
To find the equation of the original line, we begin with line $ l_1 $ and shift it three units to the right and two units up. Recall that to shift the graph of an equation three units to the right we replace $ x $ with $ x-3 $. Similarly, to shift two units up we replace $ y $ with $ y-2 $. Making these substitutions, we obtain $ y - 2 = 3(x - 3) + 7 $, which is equivalent to $ y= 3x - 14 $. Thus, $ a - b = 3 + 14 = \boxed{17} $.
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