Question

what is the equation of the line that is perpendicular to the given line and has an \\(x\\)-intercept of 6?

\\(y = -\frac{3}{4}x + 8\\)
\\(y = -\frac{3}{4}x + 6\\)
\\(y = \frac{4}{3}x - 8\\)
\\(y = \frac{4}{3}x - 6\\)

Explanation:

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<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} (x_1, y_1) &= (-4, 4) \\ (x_2, y_2) &= (4, -2) \\ m_{\text{given}} &= \frac{-2 - 4}{4 - (-4)} = \frac{-6}{8} = -\frac{3}{4} \end{aligned}$$

\]

Determine the perpendicular slope

\[

$$\begin{aligned} m_{\perp} &= -\frac{1}{m_{\text{given}}} \\ m_{\perp} &= \frac{4}{3} \end{aligned}$$

\]

Find the equation with x-intercept of 6

\[

$$\begin{aligned} (x, y) &= (6, 0) \\ y - y_1 &= m_{\perp}(x - x_1) \\ y - 0 &= \frac{4}{3}(x - 6) \\ y &= \frac{4}{3}x - 8 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = -\frac{3}{4}x + 8\)</mcq-option>
<mcq-option>(B) \(y = -\frac{3}{4}x + 6\)</mcq-option>
<mcq-correct>(C) \(y = \frac{4}{3}x - 8\)</mcq-correct>
<mcq-option>(D) \(y = \frac{4}{3}x - 6\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-4, 4]],
"properties": {
"name": "(-4, 4)",
"size": 4,
"color": "#583C87",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, -2]],
"properties": {
"name": "(4, -2)",
"size": 4,
"color": "#583C87",
"withLabel": true
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},
{
"type": "line",
"params": [[-4, 4], [4, -2]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[6, 0]],
"properties": {
"name": "(6, 0)",
"size": 4,
"color": "#F2557F",
"withLabel": true
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"type": "functiongraph",
"params": [{"js": "(4/3)*x - 8", "latex": "y = \frac{4}{3}x - 8"}, -2, 10],
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Answer:

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<reasoning>

Find the slope of the given line

\[

$$\begin{aligned} (x_1, y_1) &= (-4, 4) \\ (x_2, y_2) &= (4, -2) \\ m_{\text{given}} &= \frac{-2 - 4}{4 - (-4)} = \frac{-6}{8} = -\frac{3}{4} \end{aligned}$$

\]

Determine the perpendicular slope

\[

$$\begin{aligned} m_{\perp} &= -\frac{1}{m_{\text{given}}} \\ m_{\perp} &= \frac{4}{3} \end{aligned}$$

\]

Find the equation with x-intercept of 6

\[

$$\begin{aligned} (x, y) &= (6, 0) \\ y - y_1 &= m_{\perp}(x - x_1) \\ y - 0 &= \frac{4}{3}(x - 6) \\ y &= \frac{4}{3}x - 8 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = -\frac{3}{4}x + 8\)</mcq-option>
<mcq-option>(B) \(y = -\frac{3}{4}x + 6\)</mcq-option>
<mcq-correct>(C) \(y = \frac{4}{3}x - 8\)</mcq-correct>
<mcq-option>(D) \(y = \frac{4}{3}x - 6\)</mcq-option>
</answer>

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"elements": [
{
"type": "point",
"params": [[-4, 4]],
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"type": "point",
"params": [[4, -2]],
"properties": {
"name": "(4, -2)",
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"params": [[-4, 4], [4, -2]],
"properties": {
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{
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"params": [[6, 0]],
"properties": {
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