Question

what is the equation of the line that is perpendicular to and has the same \\(y\\)-intercept as the given line?

\\(y = \frac{1}{5}x + 1\\)
\\(y = \frac{1}{5}x + 5\\)
\\(y = 5x + 1\\)
\\(y = 5x + 5\\)

Explanation:

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<reasoning>

Find the slope and y-intercept of the given line

\[

$$\begin{aligned} &\text{Points: } (0, 1) \text{ and } (5, 0)\\ &m_{\text{given}} = \frac{0 - 1}{5 - 0} = -\frac{1}{5}\\ &b_{\text{given}} = 1 \end{aligned}$$

\]

Determine the perpendicular slope and y-intercept

\[

$$\begin{aligned} &m_{\perp} = -\frac{1}{m_{\text{given}}} = -\frac{1}{-\frac{1}{5}} = 5\\ &b_{\perp} = b_{\text{given}} = 1 \end{aligned}$$

\]

Write the equation of the perpendicular line

\[
y = 5x + 1
\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \frac{1}{5}x + 1\)</mcq-option>
<mcq-option>(B) \(y = \frac{1}{5}x + 5\)</mcq-option>
<mcq-correct>(C) \(y = 5x + 1\)</mcq-correct>
<mcq-option>(D) \(y = 5x + 5\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 1]],
"properties": {
"name": "(0, 1)",
"size": 4,
"color": "#583C87",
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},
{
"type": "point",
"params": [[5, 0]],
"properties": {
"name": "(5, 0)",
"size": 4,
"color": "#583C87",
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Answer:

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<reasoning>

Find the slope and y-intercept of the given line

\[

$$\begin{aligned} &\text{Points: } (0, 1) \text{ and } (5, 0)\\ &m_{\text{given}} = \frac{0 - 1}{5 - 0} = -\frac{1}{5}\\ &b_{\text{given}} = 1 \end{aligned}$$

\]

Determine the perpendicular slope and y-intercept

\[

$$\begin{aligned} &m_{\perp} = -\frac{1}{m_{\text{given}}} = -\frac{1}{-\frac{1}{5}} = 5\\ &b_{\perp} = b_{\text{given}} = 1 \end{aligned}$$

\]

Write the equation of the perpendicular line

\[
y = 5x + 1
\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \frac{1}{5}x + 1\)</mcq-option>
<mcq-option>(B) \(y = \frac{1}{5}x + 5\)</mcq-option>
<mcq-correct>(C) \(y = 5x + 1\)</mcq-correct>
<mcq-option>(D) \(y = 5x + 5\)</mcq-option>
</answer>

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