Answer: k = -1 +/- √769
Step-by-step explanation:
48x - ky = 11
-48x -48x
-ky = -48x + 11
[tex]\frac{-ky}{-k} = \frac{-48x}{-k} + \frac{11}{-k}[/tex]
[tex]y =\frac{48x}{k} - \frac{11}{k}[/tex]
Slope: [tex]\frac{48}{k}[/tex]
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(k + 2)x + 16y = -19
- (k + 2)x -(k + 2)x
16y = -(k + 2)x - 19
[tex]\frac{16y}{16} = -\frac{(k + 2)x}{16} - \frac{19}{16}[/tex]
[tex]y = -\frac{(k + 2)x}{16} - \frac{19}{16}[/tex]
Slope: [tex]-\frac{(k + 2)}{16}[/tex]
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[tex]\frac{48}{k}[/tex] and [tex]-\frac{(k + 2)}{16}[/tex] are perpendicular so they have opposite signs and are reciprocals of each other. When multiplied by its reciprocal, their product equals -1.
[tex]-\frac{(k + 2)}{16}[/tex] * [tex]\frac{k}{48}[/tex] = -1
[tex]\frac{(k + 2)k}{16(48)}[/tex] = 1
Cross multiply, then solve for the variable.
(k + 2)(k) = 16(48)
k² + 2k - 768 = 0
Use quadratic formula to solve:
k = -1 +/- √769
