Test for consistency and solve if possible = x + y + z = 2, 2x + 2y - z = 1, 3x + 4y + z = 9.

We are given a system of three linear equations:

\[
1. \, x + y + z = 2
\]
\[
2. \, 2x + 2y - z = 1
\]
\[
3. \, 3x + 4y + z = 9
\]

### Step 1: Check for Consistency
We need to check if these equations are consistent and have a solution. One way to do this is by solving the system using the method of elimination or substitution.

### Step 2: Solve the system using elimination

#### Subtract the first equation from the second to eliminate \( z \):
\[
(2x + 2y - z) - (x + y + z) = 1 - 2
\]
\[
x + y - 2z = -1
\]

We get:
\[
4. \, x + y - 2z = -1
\]

#### Subtract the first equation from the third to eliminate \( z \):
\[
(3x + 4y + z) - (x + y + z) = 9 - 2
\]
\[
2x + 3y = 7
\]

We get:
\[
5. \, 2x + 3y = 7
\]

### Step 3: Solve the reduced system
We now have two equations:
\[
x + y - 2z = -1 \quad \text{(Equation 4)}
\]
\[
2x + 3y = 7 \quad \text{(Equation 5)}
\]

#### Solve Equation 5 for \( x \):
\[
2x + 3y = 7 \quad \Rightarrow \quad x = \frac{7 - 3y}{2}
\]

#### Substitute \( x \) in Equation 4:
\[
\left(\frac{7 - 3y}{2}\right) + y - 2z = -1
\]
Multiply the whole equation by 2 to eliminate the fraction:
\[
(7 - 3y) + 2y - 4z = -2
\]
Simplify:
\[
7 - y - 4z = -2
\]
\[
y + 4z = 9 \quad \text{(Equation 6)}
\]

### Step 4: Solve Equation 6
Solve for \( y \) in terms of \( z \):
\[
y = 9 - 4z
\]

### Step 5: Substitute \( y \) into \( x \):
Substitute \( y = 9 - 4z \) into the expression for \( x \):
\[
x = \frac{7 - 3(9 - 4z)}{2}
\]
Simplify:
\[
x = \frac{7 - 27 + 12z}{2} = \frac{-20 + 12z}{2} = -10 + 6z
\]

### Step 6: Substitute \( x \) and \( y \) into the first equation:
Now substitute \( x = -10 + 6z \) and \( y = 9 - 4z \) into the first equation \( x + y + z = 2 \):
\[
(-10 + 6z) + (9 - 4z) + z = 2
\]
Simplify:
\[
-1 + 3z = 2
\]
\[
3z = 3 \quad \Rightarrow \quad z = 1
\]

### Step 7: Solve for \( x \) and \( y \):
Substitute \( z = 1 \) back into the expressions for \( x \) and \( y \):
\[
x = -10 + 6(1) = -4
\]
\[
y = 9 - 4(1) = 5
\]

### Final Solution:
The solution to the system is:
\[
x = -4, \quad y = 5, \quad z = 1
\]

The system is consistent and the solution is:
\[
(x, y, z) = (-4, 5, 1)
\]

Would you like further clarification or details on any steps?

---

**Related Questions:**
1. How do you determine if a system of linear equations is inconsistent?
2. What is the geometric interpretation of the solution to a system of three linear equations?
3. Can matrix methods (like Gaussian elimination) be used to solve this system more efficiently?
4. What is the general form of a system of linear equations in three variables?
5. How can substitution be used effectively in solving more complex systems?

**Tip:** Always check your solution by substituting the values back into the original equations to verify correctness!

Test for consistency and solve if possible: x + y + z = 2, 2x + 2y - z = 1, 3x + 4y + z = 9.

Learn how to solve and check the consistency of the system of linear equations: x + y + z = 2, 2x + 2y - z = 1, and 3x + 4y + z = 9. This problem involves algebraic concepts such as substitution, elimination, and solving systems of equations in three variables. Suitable for Grades 10-12.

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