Question

consider the line $y = 5x - 7$. find the equation of the line that is perpendicular to this line and passes through the point $(2, 2)$. find the equation of the line that is parallel to this line and passes through the point $(2, 2)$. note that the aleks graphing calculator may be helpful in checking your answer. equation of perpendicular line: equation of parallel line:

Explanation:

Equation of the Perpendicular Line

Step 1: Find the slope of the given line

The given line is \( y = 5x - 7 \), which is in slope - intercept form \( y=mx + b \) where \( m \) is the slope. So the slope of the given line \( m_1=5 \).

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 \( m_1 = 5 \), we get \( 5\times m_2=-1 \), so \( m_2=-\frac{1}{5} \).

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 \( (x_1,y_1)=(2,2) \) and \( m =-\frac{1}{5} \).
Substituting the values, we have \( y - 2=-\frac{1}{5}(x - 2) \).

Step 4: Convert to slope - intercept form

\( y-2=-\frac{1}{5}x+\frac{2}{5} \)
\( y=-\frac{1}{5}x+\frac{2}{5}+2 \)
\( y=-\frac{1}{5}x+\frac{2 + 10}{5} \)
\( y=-\frac{1}{5}x+\frac{12}{5} \)

Equation of the Parallel Line

Step 1: Find the slope of the parallel line

If two lines are parallel, they have the same slope. Since the slope of the given line \( y = 5x-7 \) is \( m = 5 \), the slope of the parallel line \( m_3 = 5 \).

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 \( (x_1,y_1)=(2,2) \) and \( m = 5 \).
We get \( y - 2=5(x - 2) \).

Step 3: Convert to slope - intercept form

\( y-2 = 5x-10 \)
\( y=5x-10 + 2 \)
\( y=5x-8 \)

Answer:

Equation of perpendicular line: \( y=-\frac{1}{5}x+\frac{12}{5} \)
Equation of parallel line: \( y = 5x-8 \)