Question

question 9 sec 5.2 day 2 - writing equations in point - slope form this question has two parts. first, answer part a. then, answer part b. part a correct! 2 out of 2 points. consider the graphs of the following equations. $y = -2x$, $2y = x$, $4y = 2x + 4$ a. which equations are parallel? explain your reasoning. $2y = x$ and $4y = 2x + 4$ are parallel because they have the same slope. part b incorrect 1 try left. try once more b. which equations are perpendicular? explain your reasoning. $y = -2x$ and $2y = x$ are perpendicular because they have the same slope. select choice are perpendicular because they have select choice.

Explanation:

Step1: Find slopes of each equation

• For \( y = -2x \), slope \( m_1 = -2 \) (in \( y = mx + b \) form, \( m \) is slope).
• For \( 2y = x \), rewrite as \( y=\frac{1}{2}x \), so slope \( m_2=\frac{1}{2} \).
• For \( 4y = 2x + 4 \), rewrite as \( y=\frac{2}{4}x + 1=\frac{1}{2}x + 1 \), so slope \( m_3=\frac{1}{2} \).

Step2: Check perpendicular slopes

Two lines are perpendicular if the product of their slopes is \( -1 \) (i.e., \( m_1 \times m_2=-1 \)).

• Check \( y = -2x \) (slope \( -2 \)) and \( 2y = x \) (slope \( \frac{1}{2} \)): \( (-2)\times\frac{1}{2}=-1 \). So their slopes multiply to \( -1 \).
• The other pair (\( 2y = x \) and \( 4y = 2x + 4 \)) have the same slope (\( \frac{1}{2} \)), so they are parallel, not perpendicular.

Answer:

\( y = -2x \) and \( 2y = x \) are perpendicular because the product of their slopes (\( -2 \) and \( \frac{1}{2} \)) is \( -1 \) (since \( -2\times\frac{1}{2}=-1 \), and two lines are perpendicular if the product of their slopes is \( -1 \)).