Question

find the equation of the line that passes through the point (7, -8) and is perpendicular to the line $y = \frac{1}{8}x - 2$. write the equation using the slope-intercept form.

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = \frac{1}{8}x - 2 \) is \( m_1=\frac{1}{8} \). For two perpendicular lines, the product of their slopes is \( - 1 \), so if the slope of the required line is \( m_2 \), then \( m_1\times m_2=-1 \). Substituting \( m_1 = \frac{1}{8} \), we get \( \frac{1}{8}\times m_2=-1 \), so \( m_2=-8 \).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(7,-8) \) and \( m = - 8 \). Substituting these values, we have \( y-(-8)=-8(x - 7) \). Simplifying this, we get \( y + 8=-8x+56 \).

Step3: Convert to slope - intercept form

Subtract 8 from both sides of the equation \( y + 8=-8x + 56 \). So \( y=-8x+56 - 8 \), which simplifies to \( y=-8x + 48 \).

Answer:

\( y=-8x + 48 \)