Question

write an equation in slope-intercept form for the line that passes through the given point and is perpendicular to the graph of the equation. (9, 12); y = 13x - 4

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = 13x - 4 \) is \( m_1 = 13 \). For a line perpendicular to it, the slope \( m_2 \) satisfies \( m_1 \times m_2=- 1 \). So \( 13\times m_2=-1\), which gives \( m_2 =-\frac{1}{13} \).

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 \( (x_1,y_1)=(9,12) \) and \( m =-\frac{1}{13} \). Substituting these values, we get \( y - 12=-\frac{1}{13}(x - 9) \).

Step3: Convert to slope - intercept form

Expand the right - hand side: \( y - 12=-\frac{1}{13}x+\frac{9}{13} \). Then add 12 to both sides. We know that \( 12=\frac{156}{13} \), so \( y=-\frac{1}{13}x+\frac{9}{13}+\frac{156}{13} \). Combining the fractions, we get \( y =-\frac{1}{13}x+\frac{9 + 156}{13}=-\frac{1}{13}x+\frac{165}{13} \).

Answer:

\( y=-\frac{1}{13}x+\frac{165}{13} \)