Question

5 the graphs of lines c and d are shown on the grid.
which system of equations is best represented by this graph?
a \\(\

$$\begin{aligned} y &= 2x + 6 \\\\ y &= -4x - 12 \\end{aligned}$$

\\)
b \\(\

$$\begin{aligned} y &= \\frac{1}{2}x + 6 \\\\ y &= -4x - 12 \\end{aligned}$$

\\)
c \\(\

$$\begin{aligned} y &= 2x + 6 \\\\ y &= -4x - 3 \\end{aligned}$$

\\)
d \\(\

$$\begin{aligned} y &= \\frac{1}{2}x + 6 \\\\ y &= -4x - 3 \\end{aligned}$$

\\)

Explanation:

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<reasoning>

Analyze line d

Using the Slope-Intercept Form and Systems of Linear Equations knowledge points
\[

$$\begin{aligned} &\text{Points on line } d: (-6, 3) \text{ and } (2, 7)\\ &m_d = \frac{7 - 3}{2 - (-6)} = \frac{4}{8} = \frac{1}{2}\\ &y - 7 = \frac{1}{2}(x - 2) \implies y = \frac{1}{2}x + 6 \end{aligned}$$

\]

Analyze line c

Using the Slope-Intercept Form and Systems of Linear Equations knowledge points
\[

$$\begin{aligned} &\text{Points on line } c: (-3, 0) \text{ and } (-2, -4)\\ &m_c = \frac{-4 - 0}{-2 - (-3)} = \frac{-4}{1} = -4\\ &y - 0 = -4(x - (-3)) \implies y = -4x - 12 \end{aligned}$$

\]

Match with options

Using the Systems of Linear Equations knowledge point
\[

$$\begin{aligned} &\text{System of equations:}\\ &\begin{cases} y = \frac{1}{2}x + 6\\ y = -4x - 12 \end{cases} \end{aligned}$$

\]
This matches option B.
</reasoning>

<answer>
<mcq-option>A \(

$$\begin{aligned} y &= 2x + 6 \\ y &= -4x - 12 \end{aligned}$$

\)</mcq-option>
<mcq-correct>B \(

$$\begin{aligned} y &= \frac{1}{2}x + 6 \\ y &= -4x - 12 \end{aligned}$$

\)</mcq-correct>
<mcq-option>C \(

$$\begin{aligned} y &= 2x + 6 \\ y &= -4x - 3 \end{aligned}$$

\)</mcq-option>
<mcq-option>D \(

$$\begin{aligned} y &= \frac{1}{2}x + 6 \\ y &= -4x - 3 \end{aligned}$$

\)</mcq-option>
</answer>

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</post_analysis>

Answer:

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<reasoning>

Analyze line d

Using the Slope-Intercept Form and Systems of Linear Equations knowledge points
\[

$$\begin{aligned} &\text{Points on line } d: (-6, 3) \text{ and } (2, 7)\\ &m_d = \frac{7 - 3}{2 - (-6)} = \frac{4}{8} = \frac{1}{2}\\ &y - 7 = \frac{1}{2}(x - 2) \implies y = \frac{1}{2}x + 6 \end{aligned}$$

\]

Analyze line c

Using the Slope-Intercept Form and Systems of Linear Equations knowledge points
\[

$$\begin{aligned} &\text{Points on line } c: (-3, 0) \text{ and } (-2, -4)\\ &m_c = \frac{-4 - 0}{-2 - (-3)} = \frac{-4}{1} = -4\\ &y - 0 = -4(x - (-3)) \implies y = -4x - 12 \end{aligned}$$

\]

Match with options

Using the Systems of Linear Equations knowledge point
\[

$$\begin{aligned} &\text{System of equations:}\\ &\begin{cases} y = \frac{1}{2}x + 6\\ y = -4x - 12 \end{cases} \end{aligned}$$

\]
This matches option B.
</reasoning>

<answer>
<mcq-option>A \(

$$\begin{aligned} y &= 2x + 6 \\ y &= -4x - 12 \end{aligned}$$

\)</mcq-option>
<mcq-correct>B \(

$$\begin{aligned} y &= \frac{1}{2}x + 6 \\ y &= -4x - 12 \end{aligned}$$

\)</mcq-correct>
<mcq-option>C \(

$$\begin{aligned} y &= 2x + 6 \\ y &= -4x - 3 \end{aligned}$$

\)</mcq-option>
<mcq-option>D \(

$$\begin{aligned} y &= \frac{1}{2}x + 6 \\ y &= -4x - 3 \end{aligned}$$

\)</mcq-option>
</answer>

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