To solve the system of equations:

1. \( x + y + z = 6 \)
2. \( x - y + 2z = 5 \)
3. \( x - y - 3z = -10 \)

We can use the method of elimination or substitution. Here, we'll use elimination:

First, subtract the second equation from the third to eliminate \( y \):
\[ (x - y - 3z) - (x - y + 2z) = -10 - 5 \]
\[ -3z - 2z = -15 \]
\[ -5z = -15 \]
\[ z = 3 \]

Now that we have \( z \), substitute \( z = 3 \) into the first and second equations:

1. \( x + y + 3 = 6 \)
\[ x + y = 3 \]

2. \( x - y + 2(3) = 5 \)
\[ x - y + 6 = 5 \]
\[ x - y = -1 \]

We now have a system of two equations with two variables:
1. \( x + y = 3 \)
2. \( x - y = -1 \)

Add these two equations to eliminate \( y \):
\[ (x + y) + (x - y) = 3 + (-1) \]
\[ 2x = 2 \]
\[ x = 1 \]

Substitute \( x = 1 \) back into the first of the two equations:
\[ 1 + y = 3 \]
\[ y = 2 \]

So, the solution to the system is:
\[ x = 1 \]
\[ y = 2 \]
\[ z = 3 \]

Would you like more details or have any questions?

Here are 5 questions you might want to ask:
1. How can I verify the solution to this system of equations?
2. Can you show me how to solve this system using matrix methods?
3. What are some common methods to solve systems of linear equations?
4. How can I solve systems of equations with more than three variables?
5. Can you explain the graphical interpretation of this solution?

**Tip:** Always check your solution by substituting the values back into the original equations to ensure they satisfy all equations.

Learn how to solve the system of linear equations x+y+z=6, x-y+2z=5, and x-y-3z=-10 using elimination and substitution methods. This problem involves solving simultaneous equations with three variables, suitable for high school students.

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