Question

1. *part a: lines m and n are perpendicular to each other and m passes through the point (-2,4). the equation of line n is $y = \frac{1}{3}x - 4$. write the equation of line m.\

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part b: write the equation of a line parallel to line $y = \frac{3}{4}x - 6$, that passes through the point (4, -2).\
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Explanation:

Part A

Step1: Find slope of line m

For perpendicular lines, slopes are negative reciprocals. Line n has slope \( \frac{1}{3} \), so line m's slope \( m = -3 \).

Step2: Use point - slope form

Point - slope formula: \( y - y_1 = m(x - x_1) \), with \( (x_1,y_1)=(-2,4) \) and \( m = - 3 \).
\( y - 4=-3(x + 2) \)

Step3: Simplify to slope - intercept form

\( y - 4=-3x-6 \)
\( y=-3x - 2 \)

Step1: Determine slope of new line

Parallel lines have equal slopes. Given line has slope \( \frac{3}{4} \), so new line slope \( m=\frac{3}{4} \).

Step2: Use point - slope form

Point - slope formula: \( y - y_1 = m(x - x_1) \), with \( (x_1,y_1)=(4,-2) \) and \( m=\frac{3}{4} \).
\( y + 2=\frac{3}{4}(x - 4) \)

Step3: Simplify to slope - intercept form

\( y + 2=\frac{3}{4}x-3 \)
\( y=\frac{3}{4}x-5 \)

Answer:

\( y = - 3x-2 \)

Part B