https://file.aiquickdraw.com/mathbot/file/17262889950202shq9p6y.jpg

The equation in the image involves two 2x2 matrices being added together and set equal to another matrix. The equation is as follows:

\[
\begin{bmatrix}
x & 3 \\
z & 2
\end{bmatrix}
+ 
\begin{bmatrix}
2 - y & z \\
2 - z & -x
\end{bmatrix}
=
\begin{bmatrix}
3 & 7 \\
2 & 0
\end{bmatrix}
\]

To solve for \(x\), \(y\), and \(z\), we need to add the two matrices on the left-hand side and equate the result to the matrix on the right-hand side. The matrix addition gives:

\[
\begin{bmatrix}
x + (2 - y) & 3 + z \\
z + (2 - z) & 2 + (-x)
\end{bmatrix}
=
\begin{bmatrix}
3 & 7 \\
2 & 0
\end{bmatrix}
\]

Now, let's compare the elements of the two matrices to form a system of equations:

1. \( x + (2 - y) = 3 \)
2. \( 3 + z = 7 \)
3. \( z + (2 - z) = 2 \)
4. \( 2 - x = 0 \)

### Simplifying each equation:

1. \( x + 2 - y = 3 \) → \( x - y = 1 \)
2. \( 3 + z = 7 \) → \( z = 4 \)
3. \( z + 2 - z = 2 \) → This is always true (identity).
4. \( 2 - x = 0 \) → \( x = 2 \)

Now that \( x = 2 \) and \( z = 4 \), substitute \( x = 2 \) into the first equation:

\[
2 - y = 1 \quad \Rightarrow \quad y = 1
\]

### Final values:
- \( x = 2 \)
- \( y = 1 \)
- \( z = 4 \)

Let me know if you'd like more details or clarifications. Here are 5 related questions for deeper understanding:

1. What is the general process for adding matrices?
2. How can matrix equations be used to solve systems of equations?
3. What other types of problems can be solved using matrix operations?
4. How do you check if matrix solutions are correct?
5. What are some real-world applications of matrix equations?

**Tip:** When solving matrix equations, always verify if your solution satisfies the original matrix equation by substituting the values back into the matrices.

Find x, y, z for the given matrix equation: [x, 3; z, 2] + [2 - y, z; 2 - z, -x] = [3, 7; 2, 0]

Learn how to solve the matrix equation [x, 3; z, 2] + [2 - y, z; 2 - z, -x] = [3, 7; 2, 0]. This problem covers key matrix algebra concepts, including matrix addition and solving systems of equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation