Math Problem Statement

Solve the system of equations: x + y + z = 6; x + 2y + 3z = 14; -x + y - z = -2.

Solution

The image contains a system of equations that you need to solve. The system of equations is:

1) \quad & x + y + z = 6, \\ 2) \quad & x + 2y + 3z = 14, \\ 3) \quad & -x + y - z = -2. \end{aligned}$$ Let's solve this system of linear equations using substitution or elimination. I'll proceed with elimination to find the values of $$x$$, $$y$$, and $$z$$. ### Step 1: Add equation (1) and equation (3): $$(x + y + z) + (-x + y - z) = 6 + (-2)$$ Simplifying: $$2y = 4 \implies y = 2$$ ### Step 2: Substitute $$y = 2$$ into equation (1): $$x + 2 + z = 6 \implies x + z = 4 \implies z = 4 - x$$ ### Step 3: Substitute $$y = 2$$ into equation (2): $$x + 2(2) + 3z = 14 \implies x + 4 + 3z = 14 \implies x + 3z = 10$$ Substitute $$z = 4 - x$$ from the previous step into this: $$x + 3(4 - x) = 10 \implies x + 12 - 3x = 10 \implies -2x = -2 \implies x = 1$$ ### Step 4: Substitute $$x = 1$$ into $$z = 4 - x$$: $$z = 4 - 1 = 3$$ ### Final Solution: $$x = 1, \quad y = 2, \quad z = 3$$ The solution to the system of equations is: $$x = 1, \quad y = 2, \quad z = 3.$$ Let me know if you need further details or if you have any questions. Here are five related questions: 1. How do you check if a solution to a system of equations is correct? 2. What are the methods available for solving systems of equations? 3. How does the substitution method work for solving systems of equations? 4. Can you solve this system using matrix methods like Gaussian elimination? 5. What is the geometric interpretation of solving a system of three linear equations? **Tip:** When solving systems of equations, always simplify one equation at a time to avoid confusion, and check your work by substituting back into the original equations.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations

Formulas

Addition and subtraction of equations
Substitution method

Theorems

Elimination method

Suitable Grade Level

Grades 9-12

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