⎩⎨⎧​2*x*−4*y*−*z*=53*x*−*y*=7*z*+3*y*+*x*=2​

We need to solve this system to find the values of x*x*, y*y*, and z*z*

To solve the system of linear equations:

\[
\begin{cases}
2x - 4y - z = 5 \\
3x - y = 7 \\
z + 3y + x = 2
\end{cases}
\]

We can use either substitution, elimination, or matrix methods. Here, I’ll use substitution and elimination to find the values of \(x\), \(y\), and \(z\).

### Step 1: Solve the second equation for \(y\)
The second equation is:
\[
3x - y = 7
\]
We can express \(y\) in terms of \(x\):
\[
y = 3x - 7
\]

### Step 2: Substitute \(y\) into the first and third equations
Substitute \(y = 3x - 7\) into the first and third equations.

**Substitute into the first equation:**
\[
2x - 4(3x - 7) - z = 5
\]
Simplify:
\[
2x - 12x + 28 - z = 5
\]
\[
-10x + 28 - z = 5
\]
\[
-10x - z = -23
\]
\[
z = -10x + 23
\]

**Substitute into the third equation:**
\[
z + 3(3x - 7) + x = 2
\]
Simplify:
\[
z + 9x - 21 + x = 2
\]
\[
z + 10x - 21 = 2
\]
\[
z = -10x + 23
\]

### Step 3: Compare the two expressions for \(z\)
From both the first and third equations, we derived that:
\[
z = -10x + 23
\]
This shows that the equations are consistent and have the same expression for \(z\).

### Step 4: Solution Set
Given that we have \(y = 3x - 7\) and \(z = -10x + 23\), and there’s no further restriction on \(x\), the system has infinitely many solutions. We express the general solution as:

\[
\begin{aligned}
x & = t \quad \text{(where \( t \) is any real number)} \\
y & = 3t - 7 \\
z & = -10t + 23
\end{aligned}
\]

Here, \(t\) is a parameter that can take any real value, representing the infinitely many solutions.

Would you like more details on any specific part of this process?

### Related Questions
1. What does it mean geometrically when a system of equations has infinitely many solutions?
2. How can you verify if a system of linear equations is dependent?
3. What are the differences between substitution and elimination methods?
4. Can matrix methods (like using the determinant) be applied here?
5. How would you solve the system using the elimination method instead?
6. What role does the parameter \(t\) play in the solution?
7. How does changing one coefficient affect the solution set?
8. What is the significance of a consistent system in linear algebra?

### Tip:
When solving a system with infinitely many solutions, identifying a parameter helps express the entire solution set compactly and clearly.

Learn how to solve the system of linear equations 2x - 4y - z = 5, 3x - y = 7, and z + 3y + x = 2 step-by-step. Understand the concepts of substitution and elimination methods to find the values of x, y, and z. Suitable for students in Grades 10-12.

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