Question

find the equation of the line perpendicular to $y = \frac{1}{2}x + 8$ that includes the point $(5, 3)$. give your answer in point - slope form. $y - ? = \square(x - \square)$ point - slope form: $y - y_1 = m(x - x_1)$

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = \frac{1}{2}x + 8 \) is \( m_1=\frac{1}{2} \). For a line perpendicular to it, the slope \( m \) satisfies \( m\times m_1=- 1 \). So \( m\times\frac{1}{2}=-1 \), which gives \( m=-2 \).

Step2: Identify the point \((x_1,y_1)\)

The line passes through the point \((5,3)\), so \( x_1 = 5 \) and \( y_1=3 \).

Step3: Write the point - slope form

Using the point - slope form \( y - y_1=m(x - x_1) \), substitute \( y_1 = 3 \), \( m=-2 \) and \( x_1 = 5 \). We get \( y - 3=-2(x - 5) \).

Answer:

\( y - 3=-2(x - 5) \)