To solve the system of equations:

\[3x + 2y - z = 7\]
\[x - 4y + 2z = 0\]

we'll use the method of substitution or elimination. Here are the steps:

### Step 1: Express one variable in terms of the others from one equation

From the second equation:
\[x - 4y + 2z = 0\]
\[x = 4y - 2z\]

### Step 2: Substitute this expression into the other equation

Substitute \(x = 4y - 2z\) into the first equation:

\[3(4y - 2z) + 2y - z = 7\]

### Step 3: Simplify the resulting equation

\[12y - 6z + 2y - z = 7\]
\[14y - 7z = 7\]
\[2y - z = 1\]

Now we have the system:

1. \(x = 4y - 2z\)
2. \(2y - z = 1\)

### Step 4: Solve for \(z\) in terms of \(y\)

From the second simplified equation:
\[2y - z = 1\]
\[z = 2y - 1\]

### Step 5: Substitute back to find \(x\) and \(z\)

Substitute \(z = 2y - 1\) into \(x = 4y - 2z\):

\[x = 4y - 2(2y - 1)\]
\[x = 4y - 4y + 2\]
\[x = 2\]

So we have:

\[x = 2\]
\[y = y\] (free parameter)
\[z = 2y - 1\]

### Step 6: Write the general solution

The solution to the system is:

\[x = 2\]
\[y = y\]
\[z = 2y - 1\]

Thus, the general solution can be written as:
\[ (x, y, z) = (2, y, 2y - 1) \]

### Conclusion

This system has infinitely many solutions parameterized by \(y\).

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How would you solve this system using matrix methods?
2. Can you graph the solution set of this system?
3. How does parameterizing the solution help in understanding the system?
4. What are the implications of having a free variable in the solution?
5. How would the solution change if the coefficients in the equations were different?
6. How can you use the elimination method to solve this system?
7. What does it mean for a system to be consistent?
8. How can we verify that our solution is correct?

**Tip:** When solving systems of equations, always check the number of equations versus the number of variables to determine the nature of the solution (unique, infinitely many, or none).

Learn how to solve the system of equations 3x + 2y - z = 7 and x - 4y + 2z = 0 using the substitution method. This problem explores systems of linear equations and parameterization of solutions. Suitable for students in Grades 10-12.

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