To solve the system of equations:

1. \( x + y + z = 6 \) \[ \text{(Equation 1)} \]
2. \( x + 2y + 3z = 14 \) \[ \text{(Equation 2)} \]
3. \( -x + y - z = -2 \) \[ \text{(Equation 3)} \]

### Step 1: Add Equations 1 and 3
Adding Equation 1 and Equation 3:
\[
(x + y + z) + (-x + y - z) = 6 + (-2)
\]
This simplifies to:
\[
2y = 4 \quad \Rightarrow \quad y = 2
\]

### Step 2: Substitute \( y = 2 \) into Equations 1 and 2
Substitute \( y = 2 \) into Equation 1:
\[
x + 2 + z = 6 \quad \Rightarrow \quad x + z = 4 \quad \text{(Equation A)}
\]

Substitute \( y = 2 \) into Equation 2:
\[
x + 2(2) + 3z = 14 \quad \Rightarrow \quad x + 4 + 3z = 14 \quad \Rightarrow \quad x + 3z = 10 \quad \text{(Equation B)}
\]

### Step 3: Solve Equations A and B
Now, solve the system of two equations:
1. \( x + z = 4 \) (Equation A)
2. \( x + 3z = 10 \) (Equation B)

Subtract Equation A from Equation B:
\[
(x + 3z) - (x + z) = 10 - 4
\]
This simplifies to:
\[
2z = 6 \quad \Rightarrow \quad z = 3
\]

### Step 4: Substitute \( z = 3 \) into Equation A
Substitute \( z = 3 \) into Equation A:
\[
x + 3 = 4 \quad \Rightarrow \quad x = 1
\]

### Solution:
\[
x = 1, \quad y = 2, \quad z = 3
\]

Would you like further details on this solution, or do you have any questions?

### Follow-up questions:
1. How would you solve this system using matrix methods like Gaussian elimination?
2. What happens if one of the equations is slightly modified? Will the solution change?
3. Can this system of equations be solved graphically? How would that look?
4. How would you check if the solution is unique for a system of three variables?
5. Could this system have no solution or infinitely many solutions under certain conditions?

### Tip:
When solving systems of equations, always try to simplify by eliminating variables step by step to avoid confusion.

x + y + z = 6; x + 2y + 3z = 14; -x + y - z = -2

Learn how to solve the system of linear equations: x + y + z = 6, x + 2y + 3z = 14, -x + y - z = -2. This detailed step-by-step solution covers essential algebraic methods such as substitution and elimination. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation