Question

line f has an equation of $y - 1 = \frac{1}{3}(x + 3)$. line g is perpendicular to line f and passes through $(-4, 3)$. what is the equation of line g?
write the equation in slope - intercept form. write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Explanation:

Step1: Find slope of line f

The equation of line \( f \) is \( y - 1=\frac{1}{3}(x + 3) \). In point - slope form \( y - y_1=m(x - x_1) \), the slope \( m \) of line \( f \) is \( \frac{1}{3} \).

Step2: Determine slope of line g

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of line \( g \) be \( m_g \). Then \( \frac{1}{3}\times m_g=-1 \), so \( m_g=- 3 \).

Step3: Use point - slope form for line g

Line \( g \) passes through the point \( (-4,3) \). Using the point - slope form \( y - y_1=m(x - x_1) \), where \( x_1=-4 \), \( y_1 = 3 \) and \( m=-3 \), we get \( y - 3=-3(x + 4) \).

Step4: Convert to slope - intercept form

Expand the right - hand side: \( y - 3=-3x-12 \). Add 3 to both sides: \( y=-3x-12 + 3 \), so \( y=-3x-9 \).

Answer:

\( y=-3x - 9 \)