Question

find the equation of the line perpendicular to $y = \frac{1}{3}x + 7$ that includes the point $(2, 4)$. 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 \( y = \frac{1}{3}x + 7 \) is \( m_1=\frac{1}{3} \). For two perpendicular lines, the product of their slopes is \( - 1 \), so the slope \( m \) of the perpendicular line is \( m=-3 \) (since \( \frac{1}{3}\times(-3)=-1 \)).

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

The line passes through the point \((2,4)\), so \( x_1 = 2 \) and \( y_1 = 4 \).

Step3: Write the point - slope form

Using the point - slope form \( y - y_1=m(x - x_1) \), substitute \( m=-3 \), \( x_1 = 2 \) and \( y_1 = 4 \) into the formula.

Answer:

\( y - 4=-3(x - 2) \)