Question

Set up and use matrices to solve the following system of equations

Answer 1

Sure, let's solve the system of equations using matrices. First, we define the matrices:

`\[ [A] = \begin{bmatrix} 3 & 2 \\ 2 & -1 \end{bmatrix} \]`

`\[ [X] = \begin{bmatrix} x \\ y \end{bmatrix} \]`

`\[ [B] = \begin{bmatrix} 10 \\ 3 \end{bmatrix} \]`

The matrix equation
`\([A][X] = [B]\)` is then represented as:

`\[ \begin{bmatrix} 3 & 2 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 10 \\ 3 \end{bmatrix} \]`

To solve for X, we can multiply both sides by the inverse of A:

`\[ [X] = [A]^(-1)[B] \]`

Now, after finding the inverse of A, the solution is:

`\[ [X] = \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \end{bmatrix} \]`

So, the system of equations is solved, and the ordered pair x, y is 2, 4.

The complete question is probably:

Consider the system of equations:

`\[3x + 2y = 10\]`

`\[2x - y = 3\]`

Set up a matrix equation
`\([A][X] = [B]\)` using matrices for the coefficients, variables, and constants. Define each matrix explicitly. Then, solve the system of equations using matrix operations to find the values of x and y. Finally, present the solutions in the form of an ordered pair x, y. Provide all necessary steps and calculations, ensuring clarity in your matrix operations and the final solution."