Question

consider the line $y = \dfrac{2}{9}x - 8$. find the equation of the line that is perpendicular to this line and passes through the point $(9, 6)$. find the equation of the line that is parallel to this line and passes through the point $(9, 6)$. note that the aleks graphing calculator may be helpful in checking your answer. equation of perpendicular line: equation of parallel line:

Explanation:

Step1: Recall slope properties

The given line is \( y = \frac{2}{9}x - 8 \), so its slope \( m = \frac{2}{9} \).

• For perpendicular lines: The slope of a line perpendicular to a line with slope \( m \) is \( m_{\perp}=-\frac{1}{m} \). So \( m_{\perp}=-\frac{9}{2} \).
• For parallel lines: The slope of a line parallel to a line with slope \( m \) is \( m_{\parallel}=m \). So \( m_{\parallel}=\frac{2}{9} \).

Step2: Find perpendicular line equation

Use point - slope form \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(9,6) \) and \( m = -\frac{9}{2} \).
\( y - 6=-\frac{9}{2}(x - 9) \)
Expand: \( y - 6=-\frac{9}{2}x+\frac{81}{2} \)
Add 6 to both sides: \( y=-\frac{9}{2}x+\frac{81}{2}+6 \)
\( 6=\frac{12}{2} \), so \( y = -\frac{9}{2}x+\frac{81 + 12}{2}=-\frac{9}{2}x+\frac{93}{2} \)

Step3: Find parallel line equation

Use point - slope form \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(9,6) \) and \( m=\frac{2}{9} \).
\( y - 6=\frac{2}{9}(x - 9) \)
Expand: \( y - 6=\frac{2}{9}x - 2 \)
Add 6 to both sides: \( y=\frac{2}{9}x+4 \)

Answer:

Equation of perpendicular line: \( y = -\frac{9}{2}x+\frac{93}{2} \)
Equation of parallel line: \( y=\frac{2}{9}x + 4 \)