Question

consider the line $y = \frac{3}{7}x - 4$. find the equation of the line that is perpendicular to this line and passes through the point $(9, 3)$. find the equation of the line that is parallel to this line and passes through the point $(9, 3)$. equation of perpendicular line: equation of parallel line:

Explanation:

Part 1: Equation of the Perpendicular Line

Step 1: Find the slope of the given line

The given line is \( y = \frac{3}{7}x - 4 \), which is in slope - intercept form \( y=mx + b \) where \( m \) is the slope. So the slope of the given line, \( m_1=\frac{3}{7} \).

Step 2: Find the slope of the perpendicular line

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the perpendicular line be \( m_2 \). Then \( m_1\times m_2=- 1 \).
Substituting \( m_1 = \frac{3}{7} \), we get \( \frac{3}{7}\times m_2=-1 \).
Solving for \( m_2 \), we have \( m_2=-\frac{7}{3} \).

Step 3: Use the point - slope form to find the equation of the perpendicular line

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(9,3) \) and \( m = m_2=-\frac{7}{3} \).
Substituting the values, we get \( y - 3=-\frac{7}{3}(x - 9) \).
Expand the right - hand side: \( y - 3=-\frac{7}{3}x+21 \).
Add 3 to both sides: \( y=-\frac{7}{3}x + 24 \).

Part 2: Equation of the Parallel Line

Step 1: Find the slope of the parallel line

If two lines are parallel, they have the same slope. The slope of the given line is \( m_1=\frac{3}{7} \), so the slope of the parallel line, \( m_3=\frac{3}{7} \).

Step 2: Use the point - slope form to find the equation of the parallel line

Using the point - slope form \( y - y_1=m(x - x_1) \) with \( (x_1,y_1)=(9,3) \) and \( m = \frac{3}{7} \).
We get \( y - 3=\frac{3}{7}(x - 9) \).
Expand the right - hand side: \( y - 3=\frac{3}{7}x-\frac{27}{7} \).
Add 3 (or \( \frac{21}{7} \)) to both sides: \( y=\frac{3}{7}x-\frac{27}{7}+\frac{21}{7} \).
Simplify: \( y=\frac{3}{7}x-\frac{6}{7} \).

Answer:

Equation of perpendicular line: \( y = -\frac{7}{3}x + 24 \)
Equation of parallel line: \( y=\frac{3}{7}x-\frac{6}{7} \)