Question

what do the following two equations represent?

• $y = 6x - 2$
• $2x - 12y = 24$

choose 1 answer:
a the same line
b distinct parallel lines
c perpendicular lines
d intersecting, but not perpendicular lines

Explanation:

Step1: Find slope of first line

The first equation is \( y = 6x - 2 \), 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 \( 6 \).

Step2: Rewrite second equation in slope - intercept form

We have the second equation \( 2x-12y = 24 \).
First, solve for \( y \):
Subtract \( 2x \) from both sides: \( - 12y=-2x + 24 \)
Divide each term by \( - 12 \): \( y=\frac{-2x}{-12}+\frac{24}{-12} \)
Simplify: \( y=\frac{1}{6}x - 2 \)
The slope \( m_2 \) of the second line is \( \frac{1}{6} \).

Step3: Analyze the slopes

For two lines to be parallel, their slopes must be equal (\( m_1=m_2 \)), for them to be perpendicular, the product of their slopes must be \( - 1 \) (\( m_1\times m_2=-1 \)).
Here, \( m_1 = 6 \) and \( m_2=\frac{1}{6} \), \( m_1
eq m_2 \) so they are not parallel. And \( m_1\times m_2=6\times\frac{1}{6} = 1
eq - 1 \), so they are not perpendicular. Also, since the slopes are not equal, the lines are not the same line. Since the slopes are defined (not vertical/horizontal in a way that they are parallel or same), the lines must intersect. And since they are not perpendicular, they are intersecting but not perpendicular lines.

Answer:

D. Intersecting, but not perpendicular lines