Question

1. which line is perpendicular to the line with the equation $y = \frac{3}{4}x + 12$

a. $y = \frac{3}{4}x - 7$
b. $y = -\frac{4}{3}x + 4$
c. $y = \frac{4}{3}x + 2$
d. $y = -\frac{3}{4}x + 5$

Explanation:

Step1: Recall Perpendicular Slopes 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 \( y = \frac{3}{4}x + 12 \), so its slope \( m_1=\frac{3}{4} \).

Step2: Find the Perpendicular Slope

Let the slope of the perpendicular line be \( m_2 \). Using \( m_1 \times m_2=-1 \), substitute \( m_1 = \frac{3}{4} \):
\( \frac{3}{4} \times m_2=-1 \)
Solve for \( m_2 \): \( m_2=-\frac{4}{3} \) (by multiplying both sides by \( \frac{4}{3} \)).

Step3: Identify the Line with Slope \( -\frac{4}{3} \)

Check the options:

• Option A: Slope \( \frac{3}{4} \) (not \( -\frac{4}{3} \))
• Option B: Slope \( -\frac{4}{3} \) (matches)
• Option C: Slope \( \frac{4}{3} \) (not \( -\frac{4}{3} \))
• Option D: Slope \( -\frac{3}{4} \) (not \( -\frac{4}{3} \))

Answer:

B. \( y = -\frac{4}{3}x + 4 \)