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?

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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!