Question

6-42. multiple choice: which line below is perpendicular to $y = -\frac{2}{3}x + 5$? homework help a. $2x - 3y = 6$ b. $2x + 3y = 6$ c. $3x - 2y = 6$ d. $3x + 2y = 6$

Explanation:

Step1: Find slope of given line

The given line is $y = -\frac{2}{3}x + 5$, so its slope $m_1 = -\frac{2}{3}$.

Step2: Calculate perpendicular slope

Perpendicular slopes multiply to $-1$. Let $m_2$ be the perpendicular slope:
$$m_2 = \frac{-1}{m_1} = \frac{-1}{-\frac{2}{3}} = \frac{3}{2}$$

Step3: Convert options to slope-intercept form

Option A: $2x - 3y = 6$

Rearrange to $y = \frac{2}{3}x - 2$, slope $=\frac{2}{3}$

Option B: $2x + 3y = 6$

Rearrange to $y = -\frac{2}{3}x + 2$, slope $=-\frac{2}{3}$

Option C: $3x - 2y = 6$

Rearrange to $y = \frac{3}{2}x - 3$, slope $=\frac{3}{2}$

Option D: $3x + 2y = 6$

Rearrange to $y = -\frac{3}{2}x + 3$, slope $=-\frac{3}{2}$

Step4: Match to perpendicular slope

Only Option C has slope $\frac{3}{2}$, which is perpendicular.

Answer:

C. $3x - 2y = 6$