What is the solution to the system? 5x-y+z=4 x+2y-z=5 2x+3y-3z=5

To solve the system of equations:

\[
\begin{aligned}
5x - y + z &= 4 \quad \text{(Equation 1)} \\
x + 2y - z &= 5 \quad \text{(Equation 2)} \\
2x + 3y - 3z &= 5 \quad \text{(Equation 3)}
\end{aligned}
\]

We can solve this system using methods like substitution, elimination, or matrix operations (such as Gaussian elimination). Here, I'll use the elimination method:

### Step 1: Eliminate \(z\) from Equations 1 and 2
Add Equation 1 and Equation 2 to eliminate \(z\):

\[
(5x - y + z) + (x + 2y - z) = 4 + 5
\]

\[
6x + y = 9 \quad \text{(Equation 4)}
\]

### Step 2: Eliminate \(z\) from Equations 1 and 3
Multiply Equation 1 by 3 to match the coefficient of \(z\) in Equation 3:

\[
15x - 3y + 3z = 12
\]

Subtract Equation 3 from this result to eliminate \(z\):

\[
(15x - 3y + 3z) - (2x + 3y - 3z) = 12 - 5
\]

\[
13x = 7 \quad \text{(Equation 5)}
\]

### Step 3: Solve for \(x\) using Equation 5
Solve Equation 5 for \(x\):

\[
x = \frac{7}{13}
\]

### Step 4: Substitute \(x\) into Equation 4
Substitute \(x = \frac{7}{13}\) into Equation 4:

\[
6\left(\frac{7}{13}\right) + y = 9
\]

\[
\frac{42}{13} + y = 9
\]

\[
y = 9 - \frac{42}{13} = \frac{117}{13} - \frac{42}{13} = \frac{75}{13}
\]

### Step 5: Substitute \(x\) and \(y\) into one of the original equations
Substitute \(x = \frac{7}{13}\) and \(y = \frac{75}{13}\) into Equation 1:

\[
5\left(\frac{7}{13}\right) - \frac{75}{13} + z = 4
\]

\[
\frac{35}{13} - \frac{75}{13} + z = 4
\]

\[
z = 4 - \frac{-40}{13} = \frac{52}{13} + \frac{40}{13} = \frac{92}{13}
\]

### Solution:
The solution to the system is:

\[
\boxed{x = \frac{7}{13}, \, y = \frac{75}{13}, \, z = \frac{92}{13}}
\]

Would you like more details, or do you have any questions?

### 5 Related Questions:
1. How can matrix operations like Gaussian elimination be applied to solve this system?
2. What happens if the system of equations is inconsistent?
3. How can we check if the obtained solution is correct?
4. What are the possible methods to solve a system of linear equations?
5. How does the determinant of the coefficient matrix help in solving linear systems?

### Tip:
When solving systems of equations, always verify your solution by substituting the values back into the original equations.

Learn how to solve the system of equations 5x - y + z = 4, x + 2y - z = 5, 2x + 3y - 3z = 5 with a detailed step-by-step solution using elimination method. Suitable for advanced high school or college-level students studying systems of linear equations and matrix operations.

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