Question

2i/5c level 2
1 the line graphed on the grid represents the first of two equations in a system of linear equations.

the graph of the second equation in the system passes through the point \\((2, 2)\\) and has a slope of \\(\frac{3}{4}\\). which system of equations does the graph represent?

a
\\y = -\frac{3}{2}x + 1\\
\\y = \frac{3}{4}x + \frac{1}{2}\\

b
\\y = -\frac{3}{2}x + 1\\
\\y = 2x + \frac{3}{4}\\

c
\\y = -\frac{2}{3}x + 1\\
\\y = \frac{3}{4}x + 2\\

d
\\y = -\frac{2}{3}x + 1\\
\\y = \frac{3}{4}x + \frac{1}{2}\\

Explanation:

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<reasoning>

Find the equation of the graphed line

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} &\text{y-intercept: } (0, 1) \implies b = 1\\ &\text{Another point on the line: } (3, -1)\\ &m = \frac{-1 - 1}{3 - 0} = -\frac{2}{3}\\ &y = -\frac{2}{3}x + 1 \end{aligned}$$

\]

Find the equation of the second line

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} &m = \frac{3}{4},\quad \text{Point: } (2, 2)\\ &y - 2 = \frac{3}{4}(x - 2)\\ &y = \frac{3}{4}x - \frac{3}{2} + 2\\ &y = \frac{3}{4}x + \frac{1}{2} \end{aligned}$$

\]

Match with the given options

Using the Systems of Linear Equations knowledge point
\[

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

\]
This matches option D.
</reasoning>

<answer>
<mcq-option>(A) \(

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

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

$$\begin{cases} y = \frac{3}{2}x + 1 \\ y = 2x + \frac{3}{4} \end{cases}$$

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

$$\begin{cases} y = \frac{2}{3}x + 1 \\ y = \frac{3}{4}x + 2 \end{cases}$$

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

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

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

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Answer:

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<reasoning>

Find the equation of the graphed line

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} &\text{y-intercept: } (0, 1) \implies b = 1\\ &\text{Another point on the line: } (3, -1)\\ &m = \frac{-1 - 1}{3 - 0} = -\frac{2}{3}\\ &y = -\frac{2}{3}x + 1 \end{aligned}$$

\]

Find the equation of the second line

Using the Slope-Intercept Form knowledge point
\[

$$\begin{aligned} &m = \frac{3}{4},\quad \text{Point: } (2, 2)\\ &y - 2 = \frac{3}{4}(x - 2)\\ &y = \frac{3}{4}x - \frac{3}{2} + 2\\ &y = \frac{3}{4}x + \frac{1}{2} \end{aligned}$$

\]

Match with the given options

Using the Systems of Linear Equations knowledge point
\[

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

\]
This matches option D.
</reasoning>

<answer>
<mcq-option>(A) \(

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

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

$$\begin{cases} y = \frac{3}{2}x + 1 \\ y = 2x + \frac{3}{4} \end{cases}$$

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

$$\begin{cases} y = \frac{2}{3}x + 1 \\ y = \frac{3}{4}x + 2 \end{cases}$$

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

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

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

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