Question

the equation for line f can be written as: $y = \frac{9}{5}x - 2$. perpendicular to line f is line g, which passes through the point $(5, -3)$. 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 g

The slope of line \( f \) is \( \frac{9}{5} \). For perpendicular lines, the slope of line \( g \) (\( m_g \)) is the negative reciprocal: \( m_g = -\frac{5}{9} \).

Step2: Use point - slope form

Point - slope form is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(5,-3) \) and \( m = -\frac{5}{9} \).
Substitute values: \( y - (-3)=-\frac{5}{9}(x - 5) \)
Simplify: \( y + 3=-\frac{5}{9}x+\frac{25}{9} \)

Step3: Solve for y (slope - intercept form)

Subtract 3 from both sides. \( 3=\frac{27}{9} \), so \( y=-\frac{5}{9}x+\frac{25}{9}-\frac{27}{9} \)
Simplify: \( y = -\frac{5}{9}x-\frac{2}{9} \)

Answer:

\( y = -\frac{5}{9}x-\frac{2}{9} \)