Question

what is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point \\((-4, -3)\\)?

\\(y + 3 = -4(x + 4)\\)
\\(y + 3 = -\frac{1}{4}(x + 4)\\)
\\(y + 3 = \frac{1}{4}(x + 4)\\)
\\(y + 3 = 4(x + 4)\\)

Explanation:

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<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} &(x_1, y_1) = (-1, 1)\\ &(x_2, y_2) = (0, -3)\\ &m_{\text{given}} = \frac{-3 - 1}{0 - (-1)} = \frac{-4}{1} = -4 \end{aligned}$$

\]

Determine the perpendicular slope

\[

$$\begin{aligned} &m_{\perp} = -\frac{1}{m_{\text{given}}}\\ &m_{\perp} = -\frac{1}{-4} = \frac{1}{4} \end{aligned}$$

\]

Write the equation in point-slope form

\[

$$\begin{aligned} &(x_0, y_0) = (-4, -3)\\ &y - y_0 = m_{\perp}(x - x_0)\\ &y - (-3) = \frac{1}{4}(x - (-4))\\ &y + 3 = \frac{1}{4}(x + 4) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y + 3 = -4(x + 4)\)</mcq-option>
<mcq-option>(B) \(y + 3 = -\frac{1}{4}(x + 4)\)</mcq-option>
<mcq-correct>(C) \(y + 3 = \frac{1}{4}(x + 4)\)</mcq-correct>
<mcq-option>(D) \(y + 3 = 4(x + 4)\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-1, 1]],
"properties": {
"name": "(-1, 1)",
"size": 3,
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, -3]],
"properties": {
"name": "(0, -3)",
"size": 3,
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[-4, -3]],
"properties": {
"name": "(-4, -3)",
"size": 3,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "line",
"params": [[-1, 1], [0, -3]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "functiongraph",
"params": [{"js": "0.25*(x + 4) - 3", "latex": "y + 3 = \frac{1}{4}(x + 4)"}, -10, 10],
"properties": {
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"dash": 2
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Answer:

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<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} &(x_1, y_1) = (-1, 1)\\ &(x_2, y_2) = (0, -3)\\ &m_{\text{given}} = \frac{-3 - 1}{0 - (-1)} = \frac{-4}{1} = -4 \end{aligned}$$

\]

Determine the perpendicular slope

\[

$$\begin{aligned} &m_{\perp} = -\frac{1}{m_{\text{given}}}\\ &m_{\perp} = -\frac{1}{-4} = \frac{1}{4} \end{aligned}$$

\]

Write the equation in point-slope form

\[

$$\begin{aligned} &(x_0, y_0) = (-4, -3)\\ &y - y_0 = m_{\perp}(x - x_0)\\ &y - (-3) = \frac{1}{4}(x - (-4))\\ &y + 3 = \frac{1}{4}(x + 4) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y + 3 = -4(x + 4)\)</mcq-option>
<mcq-option>(B) \(y + 3 = -\frac{1}{4}(x + 4)\)</mcq-option>
<mcq-correct>(C) \(y + 3 = \frac{1}{4}(x + 4)\)</mcq-correct>
<mcq-option>(D) \(y + 3 = 4(x + 4)\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-1, 1]],
"properties": {
"name": "(-1, 1)",
"size": 3,
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, -3]],
"properties": {
"name": "(0, -3)",
"size": 3,
"color": "#583C87",
"withLabel": true
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{
"type": "point",
"params": [[-4, -3]],
"properties": {
"name": "(-4, -3)",
"size": 3,
"color": "#F2557F",
"withLabel": true
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{
"type": "line",
"params": [[-1, 1], [0, -3]],
"properties": {
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