Question

this series of problems will build your fluency with recognizing relationships between lines.

• use the answer key to determine that you accurately identified their relationship to each other.
• use an alternate strategy to confirm that your solution is accurate.
• if necessary, revise your work. do not erase your first attempt. complete the next attempt in a new color and explain how you changed your approach.

1. are these lines equivalent, parallel, inverses, or neither? for each question below, specify the relationship between the two lines and explain how you know.

a. ( 5y = 25x ) and ( 12y = 60x )
b. the table represents one line and the equation represents another.

( x )	( y )
6	-6
8	-9
12	-15

( y=\frac{3}{2}x )

Explanation:

Step1: Simplify the equations in part a

For the equation \(5y = 25x\), divide both sides by \(5\) to get \(y = 5x\). For the equation \(12y=60x\), divide both sides by \(12\) to get \(y = 5x\).

Step2: Determine the relationship in part a

Since both equations simplify to \(y = 5x\), they are equivalent.

Step3: Find the slope of the line in the table (part b)

Use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(4,-3)\) and \((x_2,y_2)=(6,-6)\). Then \(m=\frac{-6-(-3)}{6 - 4}=\frac{-6 + 3}{2}=\frac{-3}{2}\).

Step4: Compare the slopes (part b)

The slope of the line from the table is \(m_1=-\frac{3}{2}\), and the slope of the line \(y=\frac{3}{2}x\) is \(m_2=\frac{3}{2}\). Since \(m_1
eq m_2\) and \(m_1
eq - \frac{1}{m_2}\), the lines are neither parallel nor inverses.

Answer:

a. The lines \(5y = 25x\) and \(12y = 60x\) are equivalent. Because when simplified, both give \(y = 5x\).
b. The line from the table and \(y=\frac{3}{2}x\) are neither. The slope of the line from the table is \(-\frac{3}{2}\), and the slope of \(y=\frac{3}{2}x\) is \(\frac{3}{2}\). Since their slopes are not equal (not parallel) and not negative reciprocals (not inverses).