Find all solutions to the following linear system 3x + 4y + z = 1, 2x + 3y = 0 and 4x + 3y - z = -2.

To solve the given system of linear equations:

\[
\begin{aligned}
1) & \quad 3x + 4y + z = 1 \\
2) & \quad 2x + 3y = 0 \\
3) & \quad 4x + 3y - z = -2 \\
\end{aligned}
\]

we can use either substitution, elimination, or matrix methods. Let's proceed with the elimination method.

### Step 1: Solve Equation 2 for \(y\)
Start by solving Equation 2 for \(y\):

\[
2x + 3y = 0
\]

\[
3y = -2x
\]

\[
y = -\frac{2}{3}x
\]

### Step 2: Substitute \(y = -\frac{2}{3}x\) into Equations 1 and 3
Substitute \(y = -\frac{2}{3}x\) into Equation 1:

\[
3x + 4\left(-\frac{2}{3}x\right) + z = 1
\]

Simplify:

\[
3x - \frac{8}{3}x + z = 1
\]

To clear the fraction, multiply the equation by 3:

\[
9x - 8x + 3z = 3
\]

\[
x + 3z = 3 \quad \text{(Equation 4)}
\]

Now, substitute \(y = -\frac{2}{3}x\) into Equation 3:

\[
4x + 3\left(-\frac{2}{3}x\right) - z = -2
\]

Simplify:

\[
4x - 2x - z = -2
\]

\[
2x - z = -2 \quad \text{(Equation 5)}
\]

### Step 3: Solve the resulting system (Equations 4 and 5)
We now have two equations:

\[
\begin{aligned}
4) & \quad x + 3z = 3 \\
5) & \quad 2x - z = -2 \\
\end{aligned}
\]

Let's solve this system by substitution or elimination.

**Multiply Equation 5 by 3**:

\[
6x - 3z = -6 \quad \text{(Equation 6)}
\]

Now add Equation 4 and Equation 6:

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

\[
7x = -3
\]

\[
x = -\frac{3}{7}
\]

### Step 4: Solve for \(z\) using \(x\)
Substitute \(x = -\frac{3}{7}\) into Equation 4:

\[
-\frac{3}{7} + 3z = 3
\]

\[
3z = 3 + \frac{3}{7} = \frac{21}{7} + \frac{3}{7} = \frac{24}{7}
\]

\[
z = \frac{24}{21} = \frac{8}{7}
\]

### Step 5: Solve for \(y\) using \(x\)
Substitute \(x = -\frac{3}{7}\) into \(y = -\frac{2}{3}x\):

\[
y = -\frac{2}{3} \times -\frac{3}{7} = \frac{2}{7}
\]

### Final Solution
The solution to the system is:

\[
\boxed{\left( x, y, z \right) = \left( -\frac{3}{7}, \frac{2}{7}, \frac{8}{7} \right)}
\]

Would you like any details or have any questions?

### Related Questions
1. How do you solve a system of linear equations using the substitution method?
2. What is the elimination method for solving systems of linear equations?
3. How can matrix operations be used to solve systems of equations?
4. What are the conditions for a system of linear equations to have a unique solution?
5. What is the geometric interpretation of a system of linear equations in three variables?
6. How do you verify the solution to a system of linear equations?
7. Can a system of linear equations have infinitely many solutions? If so, under what conditions?
8. How can you solve a system of linear equations using determinants?

### Tip
When solving systems of equations, always double-check your algebraic manipulations, as small errors can lead to incorrect solutions.

Learn how to solve the system of linear equations 3x + 4y + z = 1, 2x + 3y = 0, 4x + 3y - z = -2 step-by-step with detailed explanations. This problem involves using the elimination method to find the unique solution. Suitable for high school students studying algebra and systems of linear equations.

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