Question

4.11.5 stepping stones
choose the equation of the line that is perpendicular to $y = \frac{3}{4}x + 2$.
\\(\circ\\) $y = \frac{3}{4}x + 4$
\\(\circ\\) $y = \frac{4}{3}x + 3$
\\(\circ\\) $y = \frac{-3}{4}x + 1$
\\(\circ\\) $y = \frac{-4}{3}x + 2$

Explanation:

Step1: Recall slope of perpendicular lines

For two perpendicular lines with slopes \( m_1 \) and \( m_2 \), the product \( m_1 \times m_2=- 1 \), so \( m_2 =-\frac{1}{m_1} \). The given line is \( y=\frac{3}{4}x + 2 \), so its slope \( m_1=\frac{3}{4} \).

Step2: Calculate the slope of the perpendicular line

Using \( m_2=-\frac{1}{m_1} \), substitute \( m_1 = \frac{3}{4} \). Then \( m_2=-\frac{1}{\frac{3}{4}}=-\frac{4}{3} \).

Step3: Identify the line with the calculated slope

Now check the slopes of the given options:

• Option 1: \( y = \frac{3}{4}x+4 \), slope \( \frac{3}{4}

eq-\frac{4}{3} \)

• Option 2: \( y=\frac{4}{3}x + 3 \), slope \( \frac{4}{3}

eq-\frac{4}{3} \)

• Option 3: \( y=\frac{- 3}{4}x + 1 \), slope \( -\frac{3}{4}

eq-\frac{4}{3} \)

• Option 4: \( y=\frac{-4}{3}x + 2 \), slope \( -\frac{4}{3} \), which matches the slope of the perpendicular line.

Answer:

\( y=\frac{-4}{3}x + 2 \) (the fourth option)