Question

tell whether the lines for each pair of equations are parallel, perpendicular, or neither. 8. $y = \frac{3}{2}x + 9$ $18x - 12y = 12$

Explanation:

Step1: Find slope of first line

The first equation is \( y = \frac{3}{2}x + 9 \), which is in slope - intercept form \( y=mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept). So the slope \( m_1 \) of the first line is \( \frac{3}{2} \).

Step2: Rewrite second equation in slope - intercept form

We have the second equation \( 18x-12y = 12 \).
First, we solve for \( y \):
Subtract \( 18x \) from both sides: \( - 12y=-18x + 12 \)
Then divide each term by \( - 12 \):
\( y=\frac{-18x}{-12}+\frac{12}{-12} \)
Simplify the fractions: \( y=\frac{3}{2}x-1 \)
Now, this equation is in slope - intercept form \( y = mx + b \), so the slope \( m_2 \) of the second line is \( \frac{3}{2} \).

Step3: Compare the slopes

For two lines to be parallel, their slopes must be equal (\( m_1=m_2 \)). For two lines to be perpendicular, the product of their slopes must be \( - 1 \) (i.e., \( m_1\times m_2=-1 \)).
Here, \( m_1 = \frac{3}{2} \) and \( m_2=\frac{3}{2} \), so \( m_1 = m_2 \).

Answer:

The lines are parallel.