Answer: Choice C [tex]y + 7 = -\frac{4}{5}(x-6)[/tex]
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Explanation:
The given equation [tex]y = \frac{5}{4}x-5[/tex] is in the form [tex]y = mx+b[/tex] with slope [tex]m = \frac{5}{4}[/tex]
Apply the negative reciprocal to [tex]\frac{5}{4}[/tex] to get [tex]-\frac{4}{5}[/tex]. We flipped the fraction and the sign from positive to negative. The perpendicular slope is [tex]-\frac{4}{5}[/tex]
With any two perpendicular lines, their slopes always multiply to -1. This is assuming neither line is a vertical line. In this case: [tex]\frac{5}{4} \times \frac{-4}{5} = -1[/tex]
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We'll now let [tex]m = -\frac{4}{5}[/tex] and [tex](x_1,y_1) = (6,-7)[/tex]
Plug these three pieces of information into the point-slope formula
[tex]y - y_1 = m(x - x_1)\\\\y - (-7) = -\frac{4}{5}(x - 6)\\\\y + 7 = -\frac{4}{5}(x - 6)[/tex]
This points to choice C as the final answer.
