Question

1. dax correctly solved a system of linear equations by graphing below. which system did he solve?

options:
\\(\

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

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\\(\

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

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\\(\

$$\begin{cases} y = x \\\\ x - 3y = -4 \\end{cases}$$

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\\(\

$$\begin{cases} y = -x \\\\ 3x - y = -4 \\end{cases}$$

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Explanation:

Step1: Analyze the slope of the first line

The steeper line: Let's find its equation. From the graph, when \(x = 0\), \(y = 0\) (passes through origin), and slope \(m=\frac{\Delta y}{\Delta x}=\frac{- 4-0}{2 - 0}=-2\)? Wait, no, the intersection point seems to be \((2,-2)\)? Wait, no, looking at the grid, the intersection point is at \((2, - 2)\)? Wait, no, let's check the lines. The first line (steeper) has a negative slope. Let's check the options. The first equation in options: for \(y=-x\), slope is \(- 1\), for \(y = x\) slope is \(1\). The steeper line: wait, maybe better to check the intersection point. Let's assume the intersection point is \((2,-2)\). Let's plug into each option.

Option 2: \(y=-x\) (so at \(x = 2\), \(y=-2\), which matches). Then the second equation \(x + 3y=-4\). Plug \(x = 2\), \(y=-2\): \(2+3\times(-2)=2 - 6=-4\), which works.

Check other options:

Option 1: \(y = x\), at \(x = 2\), \(y = 2\), but intersection is at \(y=-2\), so no.

Option 3: \(y = x\), same as option 1, no.

Option 4: Second equation \(3x-y=-4\), plug \(x = 2\), \(y=-2\): \(3\times2-(-2)=6 + 2=8
eq-4\), so no.

So the correct system is the second option.

Step2: Verify each option

• Option 1: \(y = x\) and \(x + 3y=-4\). Intersection of \(y = x\) with \(x + 3y=-4\) is \(x+3x=-4\Rightarrow4x=-4\Rightarrow x=-1,y=-1\), not matching the graph's intersection.
• Option 2: \(y=-x\) and \(x + 3y=-4\). Intersection: substitute \(y=-x\) into \(x + 3y=-4\): \(x+3(-x)=-4\Rightarrow x-3x=-4\Rightarrow - 2x=-4\Rightarrow x = 2,y=-2\), which matches the graph's intersection.
• Option 3: \(y = x\) and \(x-3y=-4\). Substitute \(y = x\): \(x-3x=-4\Rightarrow - 2x=-4\Rightarrow x = 2,y = 2\), not matching.
• Option 4: \(y=-x\) and \(3x-y=-4\). Substitute \(y=-x\): \(3x-(-x)=-4\Rightarrow4x=-4\Rightarrow x=-1,y = 1\), not matching.

Answer:

B.

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

(assuming the second option is labeled B, as per the options given: the second radio button option is

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

)