Question

solve the following system of equations.
\\
\

$$\begin{aligned} 8x + 5y &= -2 \\\\ -3x - 7y &= 11 \\end{aligned}$$

\\
\\x = \square\\
\\y = \square\\

Explanation:

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<reasoning>

Set up the system of equations

\[

$$\begin{aligned} (1)\quad 8x + 5y &= -2 \\ (2)\quad -3x - 7y &= 11 \end{aligned}$$

\]

Eliminate x to solve for y

Multiply equation (1) by 3 and equation (2) by 8:
\[

$$\begin{aligned} 24x + 15y &= -6 \\ -24x - 56y &= 88 \end{aligned}$$

\]
Add the two equations:
\[

$$\begin{aligned} -41y &= 82 \\ y &= -2 \end{aligned}$$

\]

Substitute y back to solve for x

Substitute \(y = -2\) into equation (1):
\[

$$\begin{aligned} 8x + 5(-2) &= -2 \\ 8x - 10 &= -2 \\ 8x &= 8 \\ x &= 1 \end{aligned}$$

\]
</reasoning>

<answer>
Solve the following system of equations.
\[

$$\begin{aligned} 8x + 5y &= -2 \\ -3x - 7y &= 11 \end{aligned}$$

\]
\(x =\) <blank>1</blank>
\(y =\) <blank>-2</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Mathematics",
"Algebra",
"Systems of Linear Equations"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Set up the system of equations

\[

$$\begin{aligned} (1)\quad 8x + 5y &= -2 \\ (2)\quad -3x - 7y &= 11 \end{aligned}$$

\]

Eliminate x to solve for y

Multiply equation (1) by 3 and equation (2) by 8:
\[

$$\begin{aligned} 24x + 15y &= -6 \\ -24x - 56y &= 88 \end{aligned}$$

\]
Add the two equations:
\[

$$\begin{aligned} -41y &= 82 \\ y &= -2 \end{aligned}$$

\]

Substitute y back to solve for x

Substitute \(y = -2\) into equation (1):
\[

$$\begin{aligned} 8x + 5(-2) &= -2 \\ 8x - 10 &= -2 \\ 8x &= 8 \\ x &= 1 \end{aligned}$$

\]
</reasoning>

<answer>
Solve the following system of equations.
\[

$$\begin{aligned} 8x + 5y &= -2 \\ -3x - 7y &= 11 \end{aligned}$$

\]
\(x =\) <blank>1</blank>
\(y =\) <blank>-2</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Mathematics",
"Algebra",
"Systems of Linear Equations"
]
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</post_analysis>