Question Find the value of $z$ given a system of linear equations involving $2 \times 2$ matrices.

Studdy Solution

STEP 1

Assumptions
1. We are given two matrices and their difference.
2. We need to find the value of $z$ from the resulting matrix.
3. Matrix subtraction is performed element-wise.

STEP 2

Write down the matrix subtraction equation given in the problem.
$\left[\begin{array}{cc} x & 2x \\ 7 & 5 \end{array}\right] - 3\left[\begin{array}{cc} -1 & w \\ y & 3 \end{array}\right] = \left[\begin{array}{cc} z & 8 \\ 1 & 2w \end{array}\right]$

STEP 3

Distribute the scalar multiplication across the second matrix.
$3\left[\begin{array}{cc} -1 & w \\ y & 3 \end{array}\right] = \left[\begin{array}{cc} 3(-1) & 3w \\ 3y & 3 \cdot 3 \end{array}\right] = \left[\begin{array}{cc} -3 & 3w \\ 3y & 9 \end{array}\right]$

STEP 4

Subtract the second matrix from the first matrix, element-wise.
$\left[\begin{array}{cc} x - (-3) & 2x - 3w \\ 7 - 3y & 5 - 9 \end{array}\right] = \left[\begin{array}{cc} z & 8 \\ 1 & 2w \end{array}\right]$

STEP 5

Simplify the subtraction for each element.
$\left[\begin{array}{cc} x + 3 & 2x - 3w \\ 7 - 3y & 5 - 9 \end{array}\right] = \left[\begin{array}{cc} z & 8 \\ 1 & 2w \end{array}\right]$

STEP 6

Now, equate the corresponding elements of the matrices to find the values of the unknowns.
For the top-left elements:
$x + 3 = z$

STEP 7

Since we are only asked to find $z$ , we can stop here and state the value of $z$ .
$z = x + 3$
However, we are not given the value of $x$ , and it is not possible to determine a numerical value for $z$ without additional information about $x$ . Therefore, $z$ is expressed in terms of $x$ .

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