Question

1. a line passes through the points \\((0, 8)\\) and \\((6, 0)\\).

a. create a graph of the line.
b. circle the equation that could not be used to represent the line.
i. \\(y = -\frac{4}{3}x + 8\\)
ii. \\(y - 4 = -\frac{4}{3}(x - 3)\\)
iii. \\(3x + 4y = 24\\)

Explanation:

<pre_analysis>
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"explain_then_answer",
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"needs_drawing": true,
"concepts_used": [
"Slope Formula",
"Linear Equations",
"Point-Slope Form",
"Slope-Intercept Form",
"Linear Graphing"
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"current_concepts": [
"Slope Formula",
"Linear Equations",
"Point-Slope Form",
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"Linear Graphing"
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}
</pre_analysis>

<reasoning>

Find the slope of the line

\[
m = \frac{0 - 8}{6 - 0} = -\frac{8}{6} = -\frac{4}{3}
\]

Determine the correct equations

\[

$$\begin{aligned} &\text{Slope-intercept form: } y = -\frac{4}{3}x + 8 \quad (\text{Matches i})\\ &\text{Point-slope form at } (3, 4): y - 4 = -\frac{4}{3}(x - 3) \quad (\text{Matches ii})\\ &\text{Standard form: } 3y = -4x + 24 \implies 4x + 3y = 24 \quad (\text{Does NOT match iii: } 3x + 4y = 24) \end{aligned}$$

\]

Identify the incorrect equation

\[
3x + 4y = 24 \text{ is incorrect because the correct standard form is } 4x + 3y = 24.
\]
</reasoning>

<answer>

Question 8a

The graph is a straight line passing through the points \((0, 8)\) and \((6, 0)\).

Question 8b

The equation that could NOT be used to represent the line is:
\[\text{iii. } 3x + 4y = 24\]
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 8]],
"properties": {
"name": "(0, 8)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[6, 0]],
"properties": {
"name": "(6, 0)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "line",
"params": [[0, 8], [6, 0]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3
}
}
],
"timestamps": [0.5]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Equations"
]
}
</post_analysis>

Answer:

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"Point-Slope Form",
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</pre_analysis>

<reasoning>

Find the slope of the line

\[
m = \frac{0 - 8}{6 - 0} = -\frac{8}{6} = -\frac{4}{3}
\]

Determine the correct equations

\[

$$\begin{aligned} &\text{Slope-intercept form: } y = -\frac{4}{3}x + 8 \quad (\text{Matches i})\\ &\text{Point-slope form at } (3, 4): y - 4 = -\frac{4}{3}(x - 3) \quad (\text{Matches ii})\\ &\text{Standard form: } 3y = -4x + 24 \implies 4x + 3y = 24 \quad (\text{Does NOT match iii: } 3x + 4y = 24) \end{aligned}$$

\]

Identify the incorrect equation

\[
3x + 4y = 24 \text{ is incorrect because the correct standard form is } 4x + 3y = 24.
\]
</reasoning>

<answer>

Question 8a

The graph is a straight line passing through the points \((0, 8)\) and \((6, 0)\).

Question 8b

The equation that could NOT be used to represent the line is:
\[\text{iii. } 3x + 4y = 24\]
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[0, 8]],
"properties": {
"name": "(0, 8)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[6, 0]],
"properties": {
"name": "(6, 0)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "line",
"params": [[0, 8], [6, 0]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 3
}
}
],
"timestamps": [0.5]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Equations"
]
}
</post_analysis>