https://file.aiquickdraw.com/mathbot/file/1727793280236ia0hc6d1.jpg

Let's solve the given matrix equation step-by-step.

The problem is as follows:
\[
\begin{bmatrix} x + y & y + 2z \\ z - 1 & x \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 2 & 1 \end{bmatrix}
\]

We can equate corresponding elements from both matrices:
1. \( x + y = 3 \)
2. \( y + 2z = 8 \)
3. \( z - 1 = 2 \)
4. \( x = 1 \)

Let's solve these equations one by one:

### Step 1: Solve \( x = 1 \)
Directly from equation 4, \( x = 1 \).

### Step 2: Substitute \( x = 1 \) into \( x + y = 3 \)
\[
1 + y = 3
\]
\[
y = 2
\]

### Step 3: Substitute \( z - 1 = 2 \)
\[
z = 3
\]

### Step 4: Verification using \( y + 2z = 8 \)
Substituting \( y = 2 \) and \( z = 3 \):
\[
2 + 2(3) = 2 + 6 = 8
\]
This is correct.

Hence, the values are:
\[
x = 1, \quad y = 2, \quad z = 3
\]

Would you like any further details or have any questions about this solution?

### Related Questions
1. What would happen if the matrix had different values? Could there still be a unique solution?
2. How can you verify if a matrix equation always has a solution?
3. What are the possible ways to solve systems of linear equations with matrices?
4. How does this problem relate to finding the inverse of a matrix?
5. What are the applications of solving matrix equations in real-world problems?

### Tip
When solving matrix equations, always start by equating corresponding elements, which makes it easier to solve step-by-step.

If [x + y, y + 2z; z - 1, x] = [3, 8; 2, 1], find the values of x, y, and z.

Learn how to solve the matrix equation [x + y, y + 2z; z - 1, x] = [3, 8; 2, 1] by equating corresponding elements. This step-by-step solution covers key concepts from linear algebra and is 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