Let's solve the problem step-by-step.
We are given two matrices that are set equal to each other:
[tex]\[
\left[\begin{array}{cc} 2x & 4 \\ -5 & 8 \end{array}\right] = \left[\begin{array}{cc} -4 & y+1 \\ z & 8 \end{array}\right]
\][/tex]
Our goal is to find the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] such that the entries of the matrices are equal. We will do this by equating the corresponding elements of the two matrices.
### Step-by-Step Solution
1. Compare the element at the first row, first column:
[tex]\[
2x = -4
\][/tex]
To solve for [tex]\( x \)[/tex], divide both sides by 2:
[tex]\[
x = \frac{-4}{2}
\][/tex]
[tex]\[
x = -2
\][/tex]
2. Compare the element at the first row, second column:
[tex]\[
4 = y + 1
\][/tex]
To isolate [tex]\( y \)[/tex], subtract 1 from both sides:
[tex]\[
4 - 1 = y
\][/tex]
[tex]\[
y = 3
\][/tex]
3. Compare the element at the second row, first column:
[tex]\[
-5 = z
\][/tex]
This directly gives:
[tex]\[
z = -5
\][/tex]
4. Compare the element at the second row, second column:
[tex]\[
8 = 8
\][/tex]
This is already satisfied and doesn't provide any new information.
### Final Values
By solving the equations, we have found the values of the variables:
[tex]\[
x = -2, \quad y = 3, \quad z = -5
\][/tex]
These values satisfy the equality of the given matrices.
