question 27 points 3

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

$x = -2, y = -7$
$x = -7, y = -2$

Question

question 27 points 3

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

$x = -2, y = -7$
$x = -7, y = -2$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Coefficient Matrix", "Matrix Inverse Formula", "Solving Matrix Equations" ], "new_concepts": [], "current_concepts": [ "Coefficient Matrix", "Matrix Inverse Formula", "Solving Matrix Equations" ] } ## 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 $$ X = A^{-1}B = \begin{bmatrix} -2 & 1 \ -3 & 1 \end{bmatrix} \begin{bmatrix} 5 \ 8 \end{bmatrix} $$ $$ X = \begin{bmatrix} (-2)(5) + (1)(8) \ (-3)(5) + (1)(8) \end{bmatrix} = \begin{bmatrix} -10 + 8 \ -15 + 8 \end{bmatrix} = \begin{bmatrix} -2 \ -7 \end{bmatrix} $$ $$ x = -2, \quad y = -7 $$

$x = -2, y = -7$

$x = -7, y = -2$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Solving Matrix Equations" ] }

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Explanation:

Set up the matrix equation

Find the inverse matrix

Solve for the variables

Answer: