Question

what is the equation of the line that is parallel to the given line and has an \\(x\\)-intercept of \\(-3\\)?

\\(y = \frac{2}{3}x + 3\\)

\\(y = \frac{2}{3}x + 2\\)

\\(y = -\frac{3}{2}x + 3\\)

\\(y = -\frac{3}{2}x + 2\\)

Explanation:

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<reasoning>

Find the slope of the given line

\[
m = \frac{1 - (-1)}{3 - 0} = \frac{2}{3}
\]

Determine the equation of the parallel line

\[

$$\begin{aligned} &m_{\text{parallel}} = \frac{2}{3}\\ &\text{Point: } (-3, 0)\\ &y - 0 = \frac{2}{3}(x - (-3))\\ &y = \frac{2}{3}x + 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \frac{2}{3}x + 3\)</mcq-option>
<mcq-correct>(B) \(y = \frac{2}{3}x + 2\)</mcq-correct>
<mcq-option>(C) \(y = -\frac{3}{2}x + 3\)</mcq-option>
<mcq-option>(D) \(y = -\frac{3}{2}x + 2\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[0, -1], [3, 1]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "Given Line",
"withLabel": false
}
},
{
"type": "point",
"params": [[0, -1]],
"properties": {
"name": "(0, -1)",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[3, 1]],
"properties": {
"name": "(3, 1)",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
},
{
"type": "line",
"params": [[-3, 0], [0, 2]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"strokeDash": 2,
"name": "Parallel Line",
"withLabel": false
}
},
{
"type": "point",
"params": [[-3, 0]],
"properties": {
"name": "(-3, 0)",
"color": "#F2557F",
"size": 3,
"withLabel": true
}
}
]
}
</plot>

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Answer:

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<reasoning>

Find the slope of the given line

\[
m = \frac{1 - (-1)}{3 - 0} = \frac{2}{3}
\]

Determine the equation of the parallel line

\[

$$\begin{aligned} &m_{\text{parallel}} = \frac{2}{3}\\ &\text{Point: } (-3, 0)\\ &y - 0 = \frac{2}{3}(x - (-3))\\ &y = \frac{2}{3}x + 2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y = \frac{2}{3}x + 3\)</mcq-option>
<mcq-correct>(B) \(y = \frac{2}{3}x + 2\)</mcq-correct>
<mcq-option>(C) \(y = -\frac{3}{2}x + 3\)</mcq-option>
<mcq-option>(D) \(y = -\frac{3}{2}x + 2\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[0, -1], [3, 1]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "Given Line",
"withLabel": false
}
},
{
"type": "point",
"params": [[0, -1]],
"properties": {
"name": "(0, -1)",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[3, 1]],
"properties": {
"name": "(3, 1)",
"color": "#8C55F2",
"size": 3,
"withLabel": true
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{
"type": "line",
"params": [[-3, 0], [0, 2]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"strokeDash": 2,
"name": "Parallel Line",
"withLabel": false
}
},
{
"type": "point",
"params": [[-3, 0]],
"properties": {
"name": "(-3, 0)",
"color": "#F2557F",
"size": 3,
"withLabel": true
}
}
]
}
</plot>

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