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The problem presented involves solving a system of three linear equations to find the values of \(x\), \(y\), and \(z\). The equations are:

\[
\begin{aligned}
1) & \quad x + 2y - 3z = 6 \\
2) & \quad 2x - y + 4z = 2 \\
3) & \quad 4x + 3y - 4z = 14
\end{aligned}
\]

Let's solve this system using the substitution or elimination method.

### Step 1: Multiply Equations for Elimination

To eliminate one variable, we'll first focus on eliminating \(x\) by manipulating the equations.

Multiply the first equation by 2 and subtract the second equation:

\[
2(x + 2y - 3z) - (2x - y + 4z) = 2(6) - 2
\]

This gives us:

\[
2x + 4y - 6z - 2x + y - 4z = 12 - 2
\]

Simplifying:

\[
5y - 10z = 10 \quad \text{(Equation 4)}
\]

Divide by 5:

\[
y - 2z = 2 \quad \text{(Equation 4')}
\]

### Step 2: Use the New Equation to Eliminate Variables from the Third Equation

Multiply the first equation by 4 and subtract the third equation:

\[
4(x + 2y - 3z) - (4x + 3y - 4z) = 4(6) - 14
\]

Simplifying:

\[
4x + 8y - 12z - 4x - 3y + 4z = 24 - 14
\]

Simplifying further:

\[
5y - 8z = 10 \quad \text{(Equation 5)}
\]

### Step 3: Solve for One Variable

Subtract Equation 4' from Equation 5 to solve for \(z\):

\[
(5y - 8z) - (5y - 10z) = 10 - 10
\]

Simplifying:

\[
2z = 0 \quad \text{so} \quad z = 0
\]

### Step 4: Substitute \(z = 0\) Back

Substitute \(z = 0\) into Equation 4':

\[
y - 2(0) = 2 \quad \Rightarrow \quad y = 2
\]

### Step 5: Solve for \(x\)

Substitute \(y = 2\) and \(z = 0\) into the first equation:

\[
x + 2(2) - 3(0) = 6 \quad \Rightarrow \quad x + 4 = 6 \quad \Rightarrow \quad x = 2
\]

### Final Solution

Thus, the solution to the system is:

\[
\boxed{x = 2, \, y = 2, \, z = 0}
\]

Would you like any further details or have any questions?

Here are some related questions you might consider:

1. How can you verify the solution by substituting back into the original equations?
2. What is the geometric interpretation of solving this system of equations?
3. How would the solution change if one equation were different?
4. What method could be used to solve the system if it was non-linear?
5. How do the determinant properties of matrices relate to solving systems of equations?

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

Find out the values of x, y, and z if, x + 2y - 3z = 6, 2x - y + 4z = 2, 4x + 3y - 4z = 14.

Learn how to solve the system of linear equations x + 2y - 3z = 6, 2x - y + 4z = 2, and 4x + 3y - 4z = 14 using the elimination and substitution methods. This problem is ideal for high school students in Grades 10-12.

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