Math Problem Statement

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

Solution

The system of equations visible in the image is:

1. \quad x + y + z &= 6 \\ 2. \quad x + 2y + 3z &= 14 \\ 3. \quad -x + y - z &= -2 \end{aligned}$$ ### Solving the system of equations: We'll use the substitution or elimination method to solve this system. #### Step 1: Eliminate one variable Let's eliminate $$x$$ by adding the first and third equations. From the first and third equations: $$(x + y + z) + (-x + y - z) = 6 + (-2)$$ Simplifies to: $$2y = 4 \quad \Rightarrow \quad y = 2$$ #### Step 2: Substitute $$y = 2$$ into the first and second equations 1. Substitute $$y = 2$$ into the first equation: $$x + 2 + z = 6 \quad \Rightarrow \quad x + z = 4 \quad \Rightarrow \quad x = 4 - z \tag{4}$$ 2. Substitute $$y = 2$$ into the second equation: $$x + 2(2) + 3z = 14 \quad \Rightarrow \quad x + 4 + 3z = 14 \quad \Rightarrow \quad x + 3z = 10 \tag{5}$$ #### Step 3: Substitute $$x = 4 - z$$ from (4) into (5) Substitute $$x = 4 - z$$ into $$x + 3z = 10$$: $$(4 - z) + 3z = 10 \quad \Rightarrow \quad 4 + 2z = 10 \quad \Rightarrow \quad 2z = 6 \quad \Rightarrow \quad z = 3$$ #### Step 4: Solve for $$x$$ Now substitute $$z = 3$$ into equation (4): $$x = 4 - 3 = 1$$ #### Final solution: The solution to the system of equations is: $$x = 1, \quad y = 2, \quad z = 3$$ Let me know if you'd like a detailed breakdown of any step, or if you have further questions! --- ### Related questions: 1. What happens if we try solving this system using matrices? 2. Can the system of equations be solved using substitution for a different variable? 3. How would the solution differ if one of the equations was non-linear? 4. How does Gaussian elimination work for this system? 5. What is the significance of having a unique solution in this case? **Tip:** When solving systems of linear equations, it's useful to try both substitution and elimination methods to see which simplifies the process more efficiently.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Substitution Method
Elimination Method

Formulas

Linear equations: ax + by + cz = d
Solution of systems by substitution
Solution of systems by elimination

Theorems

Unique solution theorem for linear systems
Linear combination and substitution

Suitable Grade Level

Grades 9-12

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