Question

a.2i/5c level 1

1 the tables of ordered pairs represent some points on the graphs of line g and h.

line g
\

$$\begin{tabular}{|c|c|c|c|} \\hline x & -3 & 3 & 6 \\\\ \\hline y & 1 & -1 & -2 \\\\ \\hline \\end{tabular}$$

line h
\

$$\begin{tabular}{|c|c|c|c|} \\hline x & -2 & -1 & 0 \\\\ \\hline y & -4 & -5 & -6 \\\\ \\hline \\end{tabular}$$

which system of equations is represented by lines g and h?

a
\\(y = \frac{1}{3}x + 4\\)
\\(y = x - 6\\)

b
\\(y = -3x - 8\\)
\\(y = -x - 6\\)

c
\\(y = 3x + 10\\)
\\(y = x - 2\\)

d
\\(y = -\frac{1}{3}x\\)
\\(y = -x - 6\\)

Explanation:

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<reasoning>

Find the equation of Line g

Using the Slope-Intercept Form and Ordered Pairs knowledge points
\[

$$\begin{aligned} &m_g = \frac{-1 - 1}{3 - (-3)} = \frac{-2}{6} = -\frac{1}{3}\\ &y - 1 = -\frac{1}{3}(x - (-3)) \implies y = -\frac{1}{3}x \end{aligned}$$

\]

Find the equation of Line h

Using the Slope-Intercept Form and Ordered Pairs knowledge points
\[

$$\begin{aligned} &m_h = \frac{-5 - (-4)}{-1 - (-2)} = \frac{-1}{1} = -1\\ &b_h = -6 \quad (\text{from } (0, -6))\\ &y = -x - 6 \end{aligned}$$

\]

Formulate the system of equations

Using the Systems of Linear Equations and Linear Equations knowledge points
\[

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

\]
</reasoning>

<answer>
<mcq-option>A \(

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

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

$$\begin{aligned} y &= -3x - 8 \\ y &= -x - 6 \end{aligned}$$

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

$$\begin{aligned} y &= 3x + 10 \\ y &= x - 2 \end{aligned}$$

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

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

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

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Answer:

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<reasoning>

Find the equation of Line g

Using the Slope-Intercept Form and Ordered Pairs knowledge points
\[

$$\begin{aligned} &m_g = \frac{-1 - 1}{3 - (-3)} = \frac{-2}{6} = -\frac{1}{3}\\ &y - 1 = -\frac{1}{3}(x - (-3)) \implies y = -\frac{1}{3}x \end{aligned}$$

\]

Find the equation of Line h

Using the Slope-Intercept Form and Ordered Pairs knowledge points
\[

$$\begin{aligned} &m_h = \frac{-5 - (-4)}{-1 - (-2)} = \frac{-1}{1} = -1\\ &b_h = -6 \quad (\text{from } (0, -6))\\ &y = -x - 6 \end{aligned}$$

\]

Formulate the system of equations

Using the Systems of Linear Equations and Linear Equations knowledge points
\[

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

\]
</reasoning>

<answer>
<mcq-option>A \(

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

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

$$\begin{aligned} y &= -3x - 8 \\ y &= -x - 6 \end{aligned}$$

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

$$\begin{aligned} y &= 3x + 10 \\ y &= x - 2 \end{aligned}$$

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

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

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

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