Question

determine if the graphs are parallel, perpendicular, or neither.
$y = \frac{7}{3}x - 9$
, $3x + 7y = 4$

Explanation:

Step1: Find slope of first line

The first line is \( y = \frac{7}{3}x - 9 \). In slope - intercept form \( y=mx + b \) (where \( m \) is the slope), the slope \( m_1=\frac{7}{3} \).

Step2: Rewrite second line in slope - intercept form

The second line is \( 3x + 7y=4 \). Solve for \( y \):
Subtract \( 3x \) from both sides: \( 7y=-3x + 4 \)
Divide both sides by 7: \( y=-\frac{3}{7}x+\frac{4}{7} \)
So the slope of the second line \( m_2 = -\frac{3}{7} \)

Step3: Check parallel or perpendicular conditions

• Parallel lines have equal slopes (\( m_1 = m_2 \)). Here \( \frac{7}{3}

eq-\frac{3}{7} \), so not parallel.

• Perpendicular lines have slopes that are negative reciprocals (\( m_1\times m_2=- 1 \)). Calculate \( m_1\times m_2=\frac{7}{3}\times(-\frac{3}{7})=-1 \)

Answer:

The graphs of the two lines are perpendicular.