Math Problem Statement

What is the solution to the following system? x + 2y + z = 9, x - y + 3z = 13, 2z = 10

Solution

The system of equations provided is:

x + 2y + z &= 9 \\ x - y + 3z &= 13 \\ 2z &= 10 \end{aligned}$$ ### Step 1: Solve for $$z$$ From the third equation: $$2z = 10 \implies z = \frac{10}{2} = 5$$ ### Step 2: Substitute $$z = 5$$ into the first and second equations Substitute $$z = 5$$ into the first equation: $$x + 2y + 5 = 9 \implies x + 2y = 9 - 5 = 4 \quad \text{(Equation 1)}$$ Substitute $$z = 5$$ into the second equation: $$x - y + 3(5) = 13 \implies x - y + 15 = 13 \implies x - y = 13 - 15 = -2 \quad \text{(Equation 2)}$$ ### Step 3: Solve the system of two equations in $$x$$ and $$y$$ We now have: $$\begin{aligned} x + 2y &= 4 \quad \text{(Equation 1)} \\ x - y &= -2 \quad \text{(Equation 2)} \end{aligned}$$ To eliminate $$x$$, subtract Equation 2 from Equation 1: $$(x + 2y) - (x - y) = 4 - (-2)$$ This simplifies to: $$x + 2y - x + y = 6 \implies 3y = 6 \implies y = \frac{6}{3} = 2$$ ### Step 4: Substitute $$y = 2$$ into Equation 2 Substitute $$y = 2$$ into the equation $$x - y = -2$$: $$x - 2 = -2 \implies x = 0$$ ### Final Solution: The solution to the system is: $$x = 0, \quad y = 2, \quad z = 5$$ This corresponds to the third option: $$\boxed{x = 0, y = 2, z = 5}$$. --- Would you like further details on this solution or have any other questions? Here are five related questions to explore: 1. How do you solve a system of linear equations using substitution? 2. What is the elimination method, and how does it work for solving systems of equations? 3. How can you solve a system of equations using matrices? 4. How do you determine if a system of equations has no solution or infinitely many solutions? 5. What are the conditions for a unique solution in a system of linear equations? **Tip:** When solving a system of equations, always start by simplifying one variable if possible, like we did with $$z$$ in this example. This can make the system easier to manage.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
System of Linear Equations
Substitution
Elimination

Formulas

Substitution: Solve one equation and substitute into the others
Elimination: Combine equations to eliminate a variable

Theorems

Basic linear system solving techniques

Suitable Grade Level

Grades 9-12

Related Recommendation

Solve a System of Three Linear Equations in Three Variables

Solve a System of Linear Equations: 2x+3y-z=9, 4x-5y+2z=1, x+y+z=9

Solve System of Equations: 2x + 4y + z = 9, -2x + 10y + 3z = -1, -2x + 7y + 3z = 5

Solving a System of Linear Equations with Three Variables

Solve the System of Linear Equations: 8x - y + 7z = -2, 3x - y + z = -9, 5x + y - 6z = -9