Question

the equation of line g is $y + 3 = -\frac{2}{3}(x - 8)$. line h is perpendicular to line g and passes through $(-1, -6)$. what is the equation of line h?
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 equation of line \( g \) is in point - slope form \( y - y_1=m(x - x_1) \), where \( m \) is the slope. For line \( g \): \( y + 3=-\frac{2}{3}(x - 8) \), the slope \( m_g=-\frac{2}{3} \).

Step2: Find slope of line h

If two lines are perpendicular, the product of their slopes is \( - 1 \), i.e., \( m_g\times m_h=-1 \). Let \( m_h \) be the slope of line \( h \). We know \( m_g =-\frac{2}{3} \), so \( -\frac{2}{3}\times m_h=-1 \). Solving for \( m_h \), we get \( m_h=\frac{3}{2} \) (by multiplying both sides by \( -\frac{3}{2} \)).

Step3: Use point - slope form for line h

Line \( h \) passes through \( (-1,-6) \) and has slope \( m_h=\frac{3}{2} \). The point - slope form is \( y - y_1=m(x - x_1) \), where \( x_1=-1 \), \( y_1 = - 6 \) and \( m=\frac{3}{2} \). Substituting these values, we get \( y-(-6)=\frac{3}{2}(x - (-1)) \), which simplifies to \( y + 6=\frac{3}{2}(x + 1) \).

Step4: Convert to slope - intercept form

Expand the right - hand side: \( y+6=\frac{3}{2}x+\frac{3}{2} \). Then subtract 6 from both sides. We know that \( 6=\frac{12}{2} \), so \( y=\frac{3}{2}x+\frac{3}{2}-\frac{12}{2} \). Combining the fractions, we get \( y=\frac{3}{2}x-\frac{9}{2} \).

Answer:

\( y=\frac{3}{2}x-\frac{9}{2} \)