https://file.aiquickdraw.com/mathbot/file/1728488789754bkdft77b.jpg

The image shows a system of linear equations in matrix form:

\[
\begin{bmatrix}
2 & 3 & 0 & 0 \\
0 & 3 & 2 & 0 \\
0 & 0 & 2 & 1 \\
2 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{bmatrix}
=
\begin{bmatrix}
1000 \\
900 \\
c \\
400
\end{bmatrix}
\]

We can rewrite this system as four individual equations:

1. \( 2x_1 + 3x_2 = 1000 \)
2. \( 3x_2 + 2x_3 = 900 \)
3. \( 2x_3 + x_4 = c \)
4. \( 2x_1 + x_4 = 400 \)

### Solving the system of equations:
#### Step 1: Solve equation 4 for \(x_1\)
\[
2x_1 + x_4 = 400 \quad \Rightarrow \quad x_1 = \frac{400 - x_4}{2}
\]

#### Step 2: Solve equation 1 for \(x_1\)
\[
2x_1 + 3x_2 = 1000
\]
Substitute \(x_1 = \frac{400 - x_4}{2}\) into this equation:
\[
2\left(\frac{400 - x_4}{2}\right) + 3x_2 = 1000 \quad \Rightarrow \quad 400 - x_4 + 3x_2 = 1000
\]
\[
3x_2 = 1000 - 400 + x_4 = 600 + x_4 \quad \Rightarrow \quad x_2 = \frac{600 + x_4}{3}
\]

#### Step 3: Solve equation 2 for \(x_3\)
\[
3x_2 + 2x_3 = 900
\]
Substitute \(x_2 = \frac{600 + x_4}{3}\) into this equation:
\[
3\left(\frac{600 + x_4}{3}\right) + 2x_3 = 900 \quad \Rightarrow \quad 600 + x_4 + 2x_3 = 900
\]
\[
2x_3 = 900 - 600 - x_4 = 300 - x_4 \quad \Rightarrow \quad x_3 = \frac{300 - x_4}{2}
\]

#### Step 4: Solve equation 3 for \(x_4\)
\[
2x_3 + x_4 = c
\]
Substitute \(x_3 = \frac{300 - x_4}{2}\) into this equation:
\[
2\left(\frac{300 - x_4}{2}\right) + x_4 = c \quad \Rightarrow \quad 300 - x_4 + x_4 = c \quad \Rightarrow \quad 300 = c
\]

Thus, \(c = 300\).

### Summary of solutions:
- \(c = 300\)
- \(x_1 = \frac{400 - x_4}{2}\)
- \(x_2 = \frac{600 + x_4}{3}\)
- \(x_3 = \frac{300 - x_4}{2}\)

If you need further details or have any questions, feel free to ask!

Here are 5 follow-up questions you could explore:
1. How would the solution change if the matrix coefficients were slightly altered?
2. What happens if we substitute the value of \(c\) into the equations and solve for the other variables?
3. How would you apply matrix inversion to solve this system?
4. How do these results behave when considering parameter variations, like a different \(c\)?
5. How can we interpret this system geometrically?

**Tip**: When solving linear systems, you can use techniques like substitution, elimination, or matrix operations (such as Gaussian elimination) to find the solution efficiently.

Solve the system of linear equations represented by the matrix equation involving four unknowns: x1, x2, x3, and x4.

This problem involves solving a system of linear equations in matrix form with four unknowns. The equations are represented by a 4x4 matrix and a vector of constants. Learn how to solve it using substitution methods and matrix operations. Suitable for Grades 10-12.

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