Question

tell whether the lines for each pair of equations are parallel, perpendicular, or neither.

1. $y = \frac{7}{4}x + 2$

$21x - 12y = 15$

Explanation:

Step1: Find slope of first line

The first equation is \( y = \frac{7}{4}x + 2 \), 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 \( \frac{7}{4} \).

Step2: Rewrite second equation in slope - intercept form

We have the second equation \( 21x-12y = 15 \).
First, solve for \( y \):
Subtract \( 21x \) from both sides: \( - 12y=-21x + 15 \).
Then divide each term by \( - 12 \):
\( y=\frac{-21x}{-12}+\frac{15}{-12} \).
Simplify the fractions: \( y=\frac{7}{4}x-\frac{5}{4} \).
Now, this equation is in slope - intercept form \( y = mx + b \), so the slope \( m_2 \) of the second line is \( \frac{7}{4} \).

Step3: Compare the slopes

Two lines are parallel if their slopes are equal (\( m_1=m_2 \)), perpendicular if the product of their slopes is \( - 1\) (\( m_1\times m_2=-1 \)), and neither otherwise.
Since \( m_1 = \frac{7}{4} \) and \( m_2=\frac{7}{4} \), we have \( m_1 = m_2 \).

Answer:

The lines are parallel.