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 \( y = 6x - 2 \) is in slope - intercept form (\( y=mx + b \)), where the slope \( m_1=6 \).

Step2: Rewrite second equation in slope - intercept form

Start with \( 2x-12y = 24 \).
Subtract \( 2x \) from both sides: \(- 12y=-2x + 24\).
Divide each term by \(-12\): \(y=\frac{-2}{-12}x+\frac{24}{-12}\), which simplifies to \(y=\frac{1}{6}x - 2\).
The slope of the second line \( m_2=\frac{1}{6}\).

Step3: Analyze the slopes

For two lines to be parallel, their slopes must be equal (\( m_1 = m_2 \)), for perpendicular lines \( m_1\times m_2=- 1\).
Here, \( m_1 = 6\), \( m_2=\frac{1}{6}\), and \( 6\times\frac{1}{6}=1
eq - 1\), so they are not perpendicular. Also, \( m_1
eq m_2\), so they are not parallel. Since they are not parallel, they must intersect. And since they are not perpendicular, they are intersecting but not perpendicular lines.

Answer:

D. Intersecting, but not perpendicular lines