Solve the system. 3x+2y+z=9 -3x-y+3z=-7 2x+z=9

Let's solve the given system of equations:

1. \(3x + 2y + z = 9\)
2. \(-3x - y + 3z = -7\)
3. \(2x + z = 9\)

### Step 1: Solve for \(z\) from the third equation

From the third equation:

\[
2x + z = 9
\]

Solve for \(z\):

\[
z = 9 - 2x
\]

### Step 2: Substitute \(z = 9 - 2x\) into the first and second equations

Substitute \(z = 9 - 2x\) into the first equation:

\[
3x + 2y + (9 - 2x) = 9
\]

Simplify:

\[
3x + 2y + 9 - 2x = 9
\]

\[
x + 2y = 0 \quad \text{(Equation 4)}
\]

Substitute \(z = 9 - 2x\) into the second equation:

\[
-3x - y + 3(9 - 2x) = -7
\]

Simplify:

\[
-3x - y + 27 - 6x = -7
\]

Combine like terms:

\[
-9x - y = -34
\]

Multiply through by \(-1\):

\[
9x + y = 34 \quad \text{(Equation 5)}
\]

### Step 3: Solve for \(x\) and \(y\)

Now, solve the system formed by Equation 4 and Equation 5:

Equation 4: \(x + 2y = 0\)

Equation 5: \(9x + y = 34\)

First, solve for \(y\) in terms of \(x\) from Equation 4:

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

Substitute \(y = -\frac{x}{2}\) into Equation 5:

\[
9x + \left(-\frac{x}{2}\right) = 34
\]

Multiply everything by 2 to clear the fraction:

\[
18x - x = 68
\]

\[
17x = 68 \implies x = 4
\]

Substitute \(x = 4\) back into \(y = -\frac{x}{2}\):

\[
y = -\frac{4}{2} = -2
\]

### Step 4: Solve for \(z\)

Substitute \(x = 4\) into \(z = 9 - 2x\):

\[
z = 9 - 2(4) = 9 - 8 = 1
\]

### Step 5: Solution

Thus, the solution to the system of equations is:

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

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What methods can be used to solve a system of equations besides substitution?
2. How can you verify the solution to a system of equations?
3. What does it mean if a system of equations has no solution?
4. How would the solution change if the system had three equations that all intersected at one point?
5. What are the advantages of using matrix operations to solve systems of equations?

**Tip:** Always substitute your solutions back into the original equations to verify their correctness.

Learn how to solve the system of linear equations 3x+2y+z=9, -3x-y+3z=-7, 2x+z=9 with a step-by-step solution. This problem involves the substitution method and matrix operations, suitable for high school students.

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