Find values for the variables so that the matrices are equal.
left bracket Start 2 By 2 Matrix 1st Row 1st Column x plus 3 2nd Column y plus 4 2nd Row 1st Column 7 2nd Column negative 5 EndMatrix right bracket equals left bracket Start 2 By 2 Matrix 1st Row 1st Column 9 2nd Column 5 2nd Row 1st Column 7 2nd Column z EndMatrix right bracket

We are given two 2x2 matrices and need to find values for \( x \), \( y \), and \( z \) such that the matrices are equal. The equation is:

\[
\begin{pmatrix} 
x + 3 & y + 4 \\
7 & -5 
\end{pmatrix}
=
\begin{pmatrix} 
9 & 5 \\
7 & z 
\end{pmatrix}
\]

For two matrices to be equal, their corresponding elements must be the same. This leads to the following system of equations:

1. \( x + 3 = 9 \)
2. \( y + 4 = 5 \)
3. \( 7 = 7 \) (This is always true, so it doesn't provide any new information.)
4. \( -5 = z \)

### Solving for the variables:

1. From \( x + 3 = 9 \), we solve for \( x \):
   \[
   x = 9 - 3 = 6
   \]

2. From \( y + 4 = 5 \), we solve for \( y \):
   \[
   y = 5 - 4 = 1
   \]

3. From \( -5 = z \), we solve for \( z \):
   \[
   z = -5
   \]

Thus, the values for the variables are:
- \( x = 6 \)
- \( y = 1 \)
- \( z = -5 \)

Let me know if you need further explanations or more problems!

### Additional Questions:
1. What is the meaning of matrix equality in linear algebra?
2. How can you use matrix operations to solve systems of equations?
3. Can a matrix equation have multiple solutions? If so, under what conditions?
4. How would you solve for a variable in a matrix equation involving more than two matrices?
5. What is the difference between scalar and matrix addition?

### Tip:
Remember that when matrices are equal, their corresponding elements must match exactly!

Find values for the variables so that the matrices are equal. left bracket Start 2 By 2 Matrix 1st Row 1st Column x plus 3 2nd Column y plus 4 2nd Row 1st Column 7 2nd Column negative 5 EndMatrix right bracket equals left bracket Start 2 By 2 Matrix 1st Row 1st Column 9 2nd Column 5 2nd Row 1st Column 7 2nd Column z EndMatrix right bracket

Learn how to solve for variables in matrix equality problems. This example involves finding the values of x, y, and z in a matrix equation. Step-by-step explanation covers matrix equality and systems of linear equations. Suitable for students in Grades 9-11.

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