olve the following linear system using Elimination method 4x + 8y + 4z = 80, 2x + y – 4z = 7 and
3x – y + 2z = 22.

To solve the given system of linear equations using the elimination method, we start with the following equations:

\[
\begin{aligned}
\text{(1)} & \quad 4x + 8y + 4z = 80, \\
\text{(2)} & \quad 2x + y - 4z = 7, \\
\text{(3)} & \quad 3x - y + 2z = 22.
\end{aligned}
\]

### Step 1: Eliminate one variable
We can eliminate one variable by manipulating these equations. Let’s start by eliminating \(y\) from equations (2) and (3).

#### Multiply equation (2) by 8 to match the coefficient of \(y\) in equation (1):
\[
8 \times (2x + y - 4z = 7) \quad \Rightarrow \quad 16x + 8y - 32z = 56 \quad \text{(4)}
\]

Now subtract equation (1) from equation (4) to eliminate \(y\):
\[
(16x + 8y - 32z) - (4x + 8y + 4z) = 56 - 80
\]
\[
12x - 36z = -24 \quad \text{(5)}
\]

#### Simplify equation (5):
\[
x - 3z = -2 \quad \Rightarrow \quad x = 3z - 2 \quad \text{(6)}
\]

### Step 2: Substitute \(x = 3z - 2\) into equation (2) to solve for \(y\)
Substitute \(x = 3z - 2\) into equation (2):
\[
2(3z - 2) + y - 4z = 7
\]
\[
6z - 4 + y - 4z = 7
\]
\[
2z - 4 + y = 7
\]
\[
y = 11 - 2z \quad \text{(7)}
\]

### Step 3: Substitute \(x = 3z - 2\) and \(y = 11 - 2z\) into equation (3)
Substitute \(x = 3z - 2\) and \(y = 11 - 2z\) into equation (3):
\[
3(3z - 2) - (11 - 2z) + 2z = 22
\]
\[
9z - 6 - 11 + 2z + 2z = 22
\]
\[
13z - 17 = 22
\]
\[
13z = 39
\]
\[
z = 3 \quad \text{(8)}
\]

### Step 4: Find \(x\) and \(y\) using \(z = 3\)
Substitute \(z = 3\) into equation (6) to find \(x\):
\[
x = 3(3) - 2 = 9 - 2 = 7
\]
Substitute \(z = 3\) into equation (7) to find \(y\):
\[
y = 11 - 2(3) = 11 - 6 = 5
\]

### Final Solution
The solution to the system is:
\[
\boxed{x = 7, \ y = 5, \ z = 3}
\]

Would you like more details on any of the steps, or do you have any questions?

### Related Questions:
1. How would the elimination method change if one equation was missing?
2. Can we solve the system using substitution, and how would the process differ?
3. What if the system had no solutions or infinitely many solutions?
4. How can we verify the solution using matrix operations?
5. What role does the determinant play in solving linear systems?
6. How does the elimination method compare with the graphical method?
7. Could you explain the Gaussian elimination method for this system?
8. How would you solve the system if one of the coefficients was a fraction?

**Tip:** Always double-check your elimination steps by substituting your final values back into the original equations to ensure they satisfy all of them.

Learn how to solve the system of linear equations 4x + 8y + 4z = 80, 2x + y - 4z = 7, and 3x - y + 2z = 22 using the elimination method. This problem involves algebraic manipulation and step-by-step elimination techniques. Suitable for high school students.

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