solve the system of equations
\\begin{cases} -5x + 8y = 400 \\ 2x + 4y = 416 \end{cases}\
by converting to a matrix equation and using the inverse of the coefficient matrix.
\x = \
\y = \

Question

Accepted Answer

# Answer:
$x = 48$
$y = 80$

# Explanation:
## Step1: Write the matrix equation
$$\begin{pmatrix} -5 & 8 \ 2 & 4 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 400 \ 416 \end{pmatrix}$$

## Step2: Find the determinant of the coefficient matrix
$$\det(A) = (-5)(4) - (8)(2) = -36$$

## Step3: Find the inverse of the coefficient matrix
$$A^{-1} = -\frac{1}{36} \begin{pmatrix} 4 & -8 \ -2 & -5 \end{pmatrix} = \begin{pmatrix} -\frac{1}{9} & \frac{2}{9} \ \frac{1}{18} & \frac{5}{36} \end{pmatrix}$$

## Step4: Multiply the inverse matrix by the constant vector
$$\begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} -\frac{1}{9} & \frac{2}{9} \ \frac{1}{18} & \frac{5}{36} \end{pmatrix} \begin{pmatrix} 400 \ 416 \end{pmatrix}$$

## Step5: Calculate the value of $x$
$$x = -\frac{400}{9} + \frac{832}{9} = \frac{432}{9} = 48$$

## Step6: Calculate the value of $y$
$$y = \frac{400}{18} + \frac{2080}{36} = \frac{800}{36} + \frac{2080}{36} = \frac{2880}{36} = 80$$

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QUESTION IMAGE

Question

Explanation:

Step1: Write the matrix equation

Step2: Find the determinant of the coefficient matrix

Step3: Find the inverse of the coefficient matrix

Step4: Multiply the inverse matrix by the constant vector

Step5: Calculate the value of \(x\)

Step6: Calculate the value of \(y\)

Answer: