2x+y+2z=
2
2, x, minus, 7, y, minus, 8, z, equals, minus, 2
2x−7y−8z=
−2
minus, 2, x, plus, 4, y, plus, 3, z, equals, 8
−2x+4y+3z=
8
Answer


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.

Solve the system of equations: 2x + y + 2z = 2, 2x - 7y - 8z = -2, -2x + 4y + 3z = 8.

This step-by-step guide shows how to solve the system of linear equations: 2x + y + 2z = 2, 2x - 7y - 8z = -2, -2x + 4y + 3z = 8. Learn how to use elimination and substitution methods to find the solution (x = 3, y = 8, z = -6). Suitable for students in Grades 9-12.

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