Question

1. are the functions $3y = 4x - 2$ and $3x + 4y = 12$ parallel, perpendicular, or neither?

options:

• perpendicular
• neither
• parallel

Explanation:

Step1: Find slope of first line

Rewrite \( 3y = 4x - 2 \) in slope - intercept form \( y=mx + b \) (where \( m \) is slope). Divide both sides by 3: \( y=\frac{4}{3}x-\frac{2}{3} \). So slope \( m_1=\frac{4}{3} \).

Step2: Find slope of second line

Rewrite \( 3x + 4y = 12 \) in slope - intercept form. Subtract \( 3x \) from both sides: \( 4y=-3x + 12 \). Divide by 4: \( y=-\frac{3}{4}x + 3 \). So slope \( m_2 =-\frac{3}{4} \).

Step3: Check for perpendicularity

Two lines are perpendicular if \( m_1\times m_2=- 1 \). Calculate \( m_1\times m_2=\frac{4}{3}\times(-\frac{3}{4})=-1 \).

Answer:

perpendicular