Question

consider the line $y = -9x + 1$. find the equation of the line that is perpendicular to this line and passes through the point $(-4, -5)$. find the equation of the line that is parallel to this line and passes through the point $(-4, -5)$. 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

For two lines, if they are parallel, their slopes are equal. If they are perpendicular, the product of their slopes is -1 (or the slope of the perpendicular line is the negative reciprocal of the original slope). The given line is \( y = -9x + 1 \), so its slope \( m_1=-9 \).

Step2: Find slope of perpendicular line

Let the slope of the perpendicular line be \( m_2 \). Since \( m_1 \times m_2=-1 \), we have \( -9\times m_2 = -1 \), so \( m_2=\frac{1}{9} \). Now use point - slope form \( y - y_1=m(x - x_1) \) where \( (x_1,y_1)=(-4,-5) \). Substitute into the formula: \( y - (-5)=\frac{1}{9}(x - (-4)) \), which simplifies to \( y + 5=\frac{1}{9}(x + 4) \). Then \( y=\frac{1}{9}x+\frac{4}{9}-5=\frac{1}{9}x+\frac{4 - 45}{9}=\frac{1}{9}x-\frac{41}{9} \).

Step3: Find slope of parallel line

The slope of the parallel line \( m_3=m_1=-9 \). Use point - slope form with \( (x_1,y_1)=(-4,-5) \): \( y - (-5)=-9(x - (-4)) \), which simplifies to \( y + 5=-9(x + 4) \). Then \( y=-9x-36 - 5=-9x - 41 \).

Answer:

Equation of perpendicular line: \( y=\frac{1}{9}x-\frac{41}{9} \)
Equation of parallel line: \( y = - 9x-41 \)