select all the equations that represent a line that is perpendicular to the line with equation y = 2x - 8.
show your work here
☑️ y = $\frac{1}{2}$x + 1
☑️ x + 2y = 5
- x - 2y = 9
○ y = -$\frac{1}{2}$x + 1
☑️ -x + 2y = -3

Question

Accepted Answer

# Explanation:
To determine which lines are perpendicular to $ y = 2x - 8 $, we first recall that the slope of a line in the form $ y = mx + b $ is $ m $. For two lines to be perpendicular, the product of their slopes must be $ -1 $. The slope of $ y = 2x - 8 $ is $ 2 $, so the slope of a perpendicular line should be $ -\frac{1}{2} $ (since $ 2 \times (-\frac{1}{2}) = -1 $).

## Step 1: Analyze $ y = \frac{1}{2}x + 1 $
The slope here is $ \frac{1}{2} $. The product of $ 2 $ and $ \frac{1}{2} $ is $ 1 $, not $ -1 $. So this line is not perpendicular. (Note: There was a mistake in the initial check; this should not be selected.)

## Step 2: Analyze $ y = -\frac{1}{2}x + 1 $
The slope is $ -\frac{1}{2} $. The product of $ 2 $ and $ -\frac{1}{2} $ is $ -1 $, so this line is perpendicular.

## Step 3: Analyze $ x + 2y = 5 $
Rewrite in slope - intercept form ($ y = mx + b $):
$$
\begin{align*}
2y&=-x + 5\
y&=-\frac{1}{2}x+\frac{5}{2}
\end{align*}
$$
The slope is $ -\frac{1}{2} $. The product of $ 2 $ and $ -\frac{1}{2} $ is $ -1 $, so this line is perpendicular.

## Step 4: Analyze $ -x + 2y = -3 $
Rewrite in slope - intercept form:
$$
\begin{align*}
2y&=x - 3\
y&=\frac{1}{2}x-\frac{3}{2}
\end{align*}
$$
The slope is $ \frac{1}{2} $. The product of $ 2 $ and $ \frac{1}{2} $ is $ 1 $, not $ -1 $. So this line is not perpendicular. (Note: There was a mistake in the initial check; this should not be selected.)

## Step 5: Analyze $ -x - 2y = 9 $
Rewrite in slope - intercept form:
$$
\begin{align*}
-2y&=x + 9\
y&=-\frac{1}{2}x-\frac{9}{2}
\end{align*}
$$
The slope is $ -\frac{1}{2} $. The product of $ 2 $ and $ -\frac{1}{2} $ is $ -1 $, so this line is perpendicular. (Note: This was not initially checked, but it should be.)

Wait, let's re - do the analysis correctly:

1. For a line to be perpendicular to $ y = 2x-8 $ (slope $ m_1 = 2 $), the slope of the perpendicular line $ m_2 $ must satisfy $ m_1\times m_2=-1$, so $ m_2=-\frac{1}{2}$.

### Equation 1: $ y=\frac{1}{2}x + 1 $
Slope $ m=\frac{1}{2}$. $ 2\times\frac{1}{2}=1
eq - 1 $. Not perpendicular.

### Equation 2: $ y =-\frac{1}{2}x + 1 $
Slope $ m =-\frac{1}{2}$. $ 2\times(-\frac{1}{2})=-1 $. Perpendicular.

### Equation 3: $ x + 2y=5 $
Solve for $ y $: $ 2y=-x + 5\Rightarrow y=-\frac{1}{2}x+\frac{5}{2} $. Slope $ m =-\frac{1}{2}$. $ 2\times(-\frac{1}{2})=-1 $. Perpendicular.

### Equation 4: $ -x + 2y=-3 $
Solve for $ y $: $ 2y=x - 3\Rightarrow y=\frac{1}{2}x-\frac{3}{2} $. Slope $ m=\frac{1}{2}$. $ 2\times\frac{1}{2}=1
eq - 1 $. Not perpendicular.

### Equation 5: $ -x - 2y=9 $
Solve for $ y $: $ - 2y=x + 9\Rightarrow y=-\frac{1}{2}x-\frac{9}{2} $. Slope $ m =-\frac{1}{2}$. $ 2\times(-\frac{1}{2})=-1 $. Perpendicular.

So the correct equations are $ y =-\frac{1}{2}x + 1 $, $ x + 2y = 5 $, and $ -x - 2y = 9 $.

# Answer:
The equations representing lines perpendicular to $ y = 2x-8 $ are:
- $ y =-\frac{1}{2}x + 1 $
- $ x + 2y = 5 $
- $ -x - 2y = 9 $

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QUESTION IMAGE

Question

Explanation:

Step 1: Analyze \( y = \frac{1}{2}x + 1 \)

Step 2: Analyze \( y = -\frac{1}{2}x + 1 \)

Step 3: Analyze \( x + 2y = 5 \)

Step 4: Analyze \( -x + 2y = -3 \)

Step 5: Analyze \( -x - 2y = 9 \)

Equation 1: \( y=\frac{1}{2}x + 1 \)

Equation 2: \( y =-\frac{1}{2}x + 1 \)

Equation 3: \( x + 2y=5 \)

Equation 4: \( -x + 2y=-3 \)

Equation 5: \( -x - 2y=9 \)

Answer: