Question

the equation of line p is $y + 2 = -\frac{8}{9}(x - 2)$. line q is perpendicular to line p and passes through (7, 7). what is the equation of line q?
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 p

The equation of line \( p \) is in point - slope form \( y - y_1=m(x - x_1) \), where \( m \) is the slope. For \( y + 2=-\frac{8}{9}(x - 2) \), the slope of line \( p \), \( m_p=-\frac{8}{9} \).

Step2: Find slope of line q

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of line \( q \) be \( m_q \). Then \( m_p\times m_q=- 1 \). Substituting \( m_p =-\frac{8}{9} \), we get \( -\frac{8}{9}\times m_q=-1 \). Solving for \( m_q \), we have \( m_q=\frac{9}{8} \) (by multiplying both sides by \( -\frac{9}{8} \)).

Step3: Use point - slope form for line q

Line \( q \) passes through the point \( (7,7) \) and has slope \( m_q=\frac{9}{8} \). The point - slope form is \( y - y_1=m(x - x_1) \), where \( x_1 = 7 \), \( y_1 = 7 \) and \( m=\frac{9}{8} \). So \( y - 7=\frac{9}{8}(x - 7) \).

Step4: Convert to slope - intercept form

Expand the right - hand side: \( y - 7=\frac{9}{8}x-\frac{63}{8} \). Then add 7 to both sides. Since \( 7=\frac{56}{8} \), we have \( y=\frac{9}{8}x-\frac{63}{8}+\frac{56}{8} \). Simplifying the constants, \( y=\frac{9}{8}x-\frac{7}{8} \).

Answer:

\( y=\frac{9}{8}x-\frac{7}{8} \)