The line x - 4y = 7 has slope. . .
x - 4y = 7
-4y = -x + 7
y = (1/4)x - 7
slope = 1/4
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Perpendicular lines have slopes that are flipped as fractions (1/4 becomes 4/1) and as +- signs (4/1 becomes -4/1).
Our new slope is -4/1 or just -4.
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So our goal is to find the slope intercept form of the line passing through (3,4) with slope -4.
Point-slope form:
y - y1 = m(x - x1)
y - 4 = -4(x - 3)
y - 4 = -4x + 12
y = -4x + 16