https://file.aiquickdraw.com/mathbot/file/1728229972576z4b7nxnt.jpg

The matrix equation given in the image is:

\[
\begin{pmatrix}
3 & -1 \\
2 & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
=
\begin{pmatrix}
5 \\
5
\end{pmatrix}
\]

This is equivalent to the system of equations:

1. \( 3x_1 - x_2 = 5 \)
2. \( 2x_1 + x_2 = 5 \)

### Step 1: Solve the system of equations

First, solve for \(x_2\) from the second equation:
\[
x_2 = 5 - 2x_1
\]

### Step 2: Substitute into the first equation

Substitute \(x_2 = 5 - 2x_1\) into the first equation:
\[
3x_1 - (5 - 2x_1) = 5
\]

Simplify:
\[
3x_1 - 5 + 2x_1 = 5
\]
\[
5x_1 - 5 = 5
\]
\[
5x_1 = 10
\]
\[
x_1 = 2
\]

### Step 3: Find \(x_2\)

Substitute \(x_1 = 2\) back into the expression for \(x_2\):
\[
x_2 = 5 - 2(2) = 5 - 4 = 1
\]

### Final Solution:
The solution to the matrix equation is:
\[
X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}
\]

Let me know if you'd like more details on any step, or if you have any questions.

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Here are five related questions:

1. How do you check if a matrix equation has a unique solution?
2. What is the determinant of the matrix in this problem, and how is it relevant?
3. How would you solve this matrix equation using matrix inversion?
4. Can this system of equations be solved using Gaussian elimination? How?
5. How can this matrix equation be represented graphically?

**Tip:** When solving systems of equations, if the determinant of the coefficient matrix is non-zero, the system has a unique solution!

Solve the matrix equation: (3 -1, 2 1) * X = (5, 5)

Learn how to solve the matrix equation (3 -1, 2 1) * X = (5, 5) using the substitution method. This step-by-step solution covers key linear algebra concepts and system of equations. Suitable for students in Grades 9-12.

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