Question

question 27 points 3

solve the system of equations using the matrix inverse method.
\\(x - y = 5\\)
\\(3x - 2y = 8\\)

\\(x = -2, y = -7\\)
\\(x = -7, y = -2\\)

Explanation:

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<reasoning>

Set up the matrix equation

Using the Coefficient Matrix knowledge point
\[
A =

$$\begin{bmatrix} 1 & -1 \\ 3 & -2 \end{bmatrix}$$

, \quad X =

$$\begin{bmatrix} x \\ y \end{bmatrix}$$

, \quad B =

$$\begin{bmatrix} 5 \\ 8 \end{bmatrix}$$

\]
\[
AX = B
\]

Find the inverse matrix

Using the Matrix Inverse Formula knowledge point
\[
\det(A) = (1)(-2) - (-1)(3) = -2 + 3 = 1
\]
\[
A^{-1} = \frac{1}{1}

$$\begin{bmatrix} -2 & 1 \\ -3 & 1 \end{bmatrix}$$

=

$$\begin{bmatrix} -2 & 1 \\ -3 & 1 \end{bmatrix}$$

\]

Solve for the variables

Using the Solving Matrix Equations knowledge point
\[
X = A^{-1}B =

$$\begin{bmatrix} -2 & 1 \\ -3 & 1 \end{bmatrix}$$

$$\begin{bmatrix} 5 \\ 8 \end{bmatrix}$$

\]
\[
X =

$$\begin{bmatrix} (-2)(5) + (1)(8) \\ (-3)(5) + (1)(8) \end{bmatrix}$$

=

$$\begin{bmatrix} -10 + 8 \\ -15 + 8 \end{bmatrix}$$

=

$$\begin{bmatrix} -2 \\ -7 \end{bmatrix}$$

\]
\[
x = -2, \quad y = -7
\]
</reasoning>

<answer>
<mcq-correct>\(x = -2, y = -7\)</mcq-correct>
<mcq-option>\(x = -7, y = -2\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Set up the matrix equation

Using the Coefficient Matrix knowledge point
\[
A =

$$\begin{bmatrix} 1 & -1 \\ 3 & -2 \end{bmatrix}$$

, \quad X =

$$\begin{bmatrix} x \\ y \end{bmatrix}$$

, \quad B =

$$\begin{bmatrix} 5 \\ 8 \end{bmatrix}$$

\]
\[
AX = B
\]

Find the inverse matrix

Using the Matrix Inverse Formula knowledge point
\[
\det(A) = (1)(-2) - (-1)(3) = -2 + 3 = 1
\]
\[
A^{-1} = \frac{1}{1}

$$\begin{bmatrix} -2 & 1 \\ -3 & 1 \end{bmatrix}$$

=

$$\begin{bmatrix} -2 & 1 \\ -3 & 1 \end{bmatrix}$$

\]

Solve for the variables

Using the Solving Matrix Equations knowledge point
\[
X = A^{-1}B =

$$\begin{bmatrix} -2 & 1 \\ -3 & 1 \end{bmatrix}$$

$$\begin{bmatrix} 5 \\ 8 \end{bmatrix}$$

\]
\[
X =

$$\begin{bmatrix} (-2)(5) + (1)(8) \\ (-3)(5) + (1)(8) \end{bmatrix}$$

=

$$\begin{bmatrix} -10 + 8 \\ -15 + 8 \end{bmatrix}$$

=

$$\begin{bmatrix} -2 \\ -7 \end{bmatrix}$$

\]
\[
x = -2, \quad y = -7
\]
</reasoning>

<answer>
<mcq-correct>\(x = -2, y = -7\)</mcq-correct>
<mcq-option>\(x = -7, y = -2\)</mcq-option>
</answer>

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