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Let's solve the system of equations step-by-step. The given system is:

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

### Step 1: Add equations (1) and (3) to eliminate \(y\)
First, let's add equations (1) and (3):

\[
(3x + y - 5z) + (x + y - 5z) = 5 + 1
\]
\[
4x - 10z = 6 \quad \text{(4)}
\]

Simplifying:

\[
2x - 5z = 3 \quad \text{(4)}
\]

### Step 2: Add equations (2) and (3) to eliminate \(y\)
Next, add equations (2) and (3):

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

### Step 3: Solve the simplified system
Now we have the simplified system of two equations:

1. \(2x - 5z = 3 \quad \text{(from step 1)}\)
2. \(3x - 2z = 4 \quad \text{(from step 2)}\)

We can solve this system by eliminating \(z\). Let's multiply the first equation by 3 and the second by 2:

\[
6x - 15z = 9 \quad \text{(6)}
\]
\[
6x - 4z = 8 \quad \text{(7)}
\]

Subtract equation (7) from (6):

\[
(6x - 15z) - (6x - 4z) = 9 - 8
\]
\[
-11z = 1
\]
\[
z = -\frac{1}{11}
\]

### Step 4: Solve for \(x\)
Now substitute \(z = -\frac{1}{11}\) into one of the simplified equations, say \(3x - 2z = 4\):

\[
3x - 2\left(-\frac{1}{11}\right) = 4
\]
\[
3x + \frac{2}{11} = 4
\]
\[
3x = 4 - \frac{2}{11}
\]
\[
3x = \frac{44}{11} - \frac{2}{11}
\]
\[
3x = \frac{42}{11}
\]
\[
x = \frac{14}{11}
\]

### Step 5: Solve for \(y\)
Finally, substitute \(x = \frac{14}{11}\) and \(z = -\frac{1}{11}\) into one of the original equations, say equation (3):

\[
x + y - 5z = 1
\]
\[
\frac{14}{11} + y - 5\left(-\frac{1}{11}\right) = 1
\]
\[
\frac{14}{11} + y + \frac{5}{11} = 1
\]
\[
y + \frac{19}{11} = 1
\]
\[
y = 1 - \frac{19}{11}
\]
\[
y = \frac{11}{11} - \frac{19}{11}
\]
\[
y = -\frac{8}{11}
\]

### Final Answer:
The solution is:

\[
\left( x, y, z \right) = \left( \frac{14}{11}, -\frac{8}{11}, -\frac{1}{11} \right)
\]

Let me know if you need further details or clarifications!

### Here are 5 questions to deepen your understanding:
1. What is the method called that involves adding or subtracting equations to eliminate a variable?
2. How can you verify the solution of a system of equations?
3. Why is it useful to multiply equations when solving systems?
4. Can this method be applied to larger systems of equations, such as 4x4 systems?
5. What alternative methods can be used to solve systems of linear equations?

### Tip:
Always double-check your final solution by substituting it back into the original equations.

Solve the system of equations: 3x + y - 5z = 5, 2x - y + 3z = 3, x + y - 5z = 1. Write your answer as an ordered triple.

Step-by-step solution for solving the system of equations 3x + y - 5z = 5, 2x - y + 3z = 3, and x + y - 5z = 1. Learn how to use the elimination method to find the values of x, y, and z. Suitable for high school students in Grades 10-12.

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