The system of equations provided is:

$1) \ \ 2x + y + 2z = 2$ $2) \ \ 2x - 7y - 8z = -2$ $3) \ \ -2x + 4y + 3z = 8$

We can solve this system of linear equations using methods like substitution, elimination, or matrix techniques such as Gaussian elimination. Let's solve it step-by-step using elimination or substitution.

Step 1: Add equations (1) and (3)

First, add equation (1) and (3) to eliminate $x$:

$(2x + y + 2z) + (-2x + 4y + 3z) = 2 + 8$ $0x + 5y + 5z = 10$ $5y + 5z = 10 \quad \text{(divide the whole equation by 5)}$ $y + z = 2 \quad \text{(Equation 4)}$

Step 2: Subtract equation (1) from (2)

Now subtract equation (1) from (2) to further reduce the system:

$(2x - 7y - 8z) - (2x + y + 2z) = -2 - 2$ $0x - 8y - 10z = -4$ $-8y - 10z = -4 \quad \text{(Equation 5)}$

Step 3: Solve system with Equations (4) and (5)

We now have the reduced system of equations:

1. $y + z = 2$
2. $-8y - 10z = -4$

Substitute $z = 2 - y$ (from equation 4) into equation (5):

$-8y - 10(2 - y) = -4$ $-8y - 20 + 10y = -4$ $2y - 20 = -4$ $2y = 16$ $y = 8$

Step 4: Solve for $z$

Now, substitute $y = 8$ into equation (4):

$8 + z = 2$ $z = 2 - 8 = -6$

Step 5: Solve for $x$

Substitute $y = 8$ and $z = -6$ into equation (1):

$2x + 8 + 2(-6) = 2$ $2x + 8 - 12 = 2$ $2x - 4 = 2$ $2x = 6$ $x = 3$

Final Solution:

The solution to the system of equations is: $x = 3, \ y = 8, \ z = -6$

Would you like further details or explanations on any step? Here are 5 related questions to explore the topic further:

1. How can you solve systems of equations using substitution?
2. What is the Gaussian elimination method for solving linear systems?
3. How can you express a system of equations as a matrix equation?
4. How do linear systems relate to vector spaces?
5. What are the conditions for a system of equations to have no solution, one solution, or infinite solutions?

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