Question

which of the following equations would be perpendicular to the equation shown?
$y = \frac{1}{3}x + 4$
graph of the line $y = \frac{1}{3}x + 4$ on a coordinate plane with grid lines from -5 to 5 on both axes
options:
a) $y = -\frac{1}{3}x + 4$
b) $y = 3x + 4$
c) $y = -3x + 4$
d) $y = -\frac{3}{4}x + 4$

Explanation:

Step1: Recall perpendicular slope rule

For two lines to be perpendicular, the product of their slopes \(m_1\) and \(m_2\) is \(-1\), i.e., \(m_1\times m_2=-1\).
The given line is \(y = \frac{1}{3}x + 4\), so its slope \(m_1=\frac{1}{3}\).

Step2: Find the slope of the perpendicular line

Let the slope of the perpendicular line be \(m_2\). Then \(\frac{1}{3}\times m_2=-1\).
Solving for \(m_2\), we get \(m_2=-1\div\frac{1}{3}=-3\).

Step3: Identify the line with slope -3

Now check the options:

• Option A: Slope is \(-\frac{1}{3}\), not -3.
• Option B: Slope is 3, not -3.
• Option C: Slope is -3, which matches.
• Option D: Slope is \(-\frac{3}{4}\), not -3.

Answer:

C. \(y = -3x + 4\)