We are given the system of equations:

$1) \ 2x + y - z = 3$ $2) \ 3x - y + 2z = 4$ $3) \ x - y - 3z = -8$

To solve this system of equations, we can use either substitution or elimination. Here, we will use the elimination method.

Step 1: Eliminate one variable

Let's first eliminate $y$ by adding and subtracting equations.

• Add equation 1 and equation 2: $(2x + y - z) + (3x - y + 2z) = 3 + 4$ This simplifies to: $5x + z = 7 \quad \text{(Equation 4)}$

• Subtract equation 2 from equation 3: $(x - y - 3z) - (3x - y + 2z) = -8 - 4$ This simplifies to: $-2x - 5z = -12 \quad \text{(Equation 5)}$

Step 2: Solve for one variable

Now we have two equations with $x$ and $z$:

$4) \ 5x + z = 7$ $5) \ -2x - 5z = -12$

• Multiply equation 4 by 5 to make the coefficient of $z$ the same: $25x + 5z = 35$ Now add this to equation 5: $(25x + 5z) + (-2x - 5z) = 35 + (-12)$ This simplifies to: $23x = 23$ So: $x = 1$

Step 3: Substitute $x = 1$ back into one of the equations

Substitute $x = 1$ into equation 4: $5(1) + z = 7$ $5 + z = 7$ $z = 2$

Step 4: Solve for $y$

Substitute $x = 1$ and $z = 2$ into equation 1: $2(1) + y - 2 = 3$ $2 + y - 2 = 3$ $y = 3$

Final Solution:

The solution to the system is: $x = 1, \quad y = 3, \quad z = 2$

Would you like further details or have any questions?

Here are 5 related questions:

1. Can you solve the system using the substitution method instead of elimination?
2. How would you solve the system if there were three variables but only two equations?
3. What happens if two equations in the system are dependent on each other?
4. Can you represent this system of equations in matrix form?
5. How do you check if a solution to a system of equations is unique?

Tip: Always check your solution by substituting the values back into the original equations.