Math Problem Statement
x+y-2z=5 / -x+2y+z=2 / 2x+3y-z=9
Solution
You have a system of three linear equations:
1. & \quad x + y - 2z = 5 \\ 2. & \quad -x + 2y + z = 2 \\ 3. & \quad 2x + 3y - z = 9 \end{aligned}$$ I will solve this system using the method of substitution, elimination, or matrix operations. Let's use the elimination method. ### Step 1: Add Equation 1 and Equation 2 Adding the first and second equations to eliminate $$x$$: $$(x + y - 2z) + (-x + 2y + z) = 5 + 2$$ $$y + 2y - 2z + z = 7$$ $$3y - z = 7 \quad \text{(Equation 4)}$$ ### Step 2: Add Equation 2 and Equation 3 Adding the second and third equations to eliminate $$z$$: $$(-x + 2y + z) + (2x + 3y - z) = 2 + 9$$ $$x + 5y = 11 \quad \text{(Equation 5)}$$ ### Step 3: Solve Equation 4 and Equation 5 Now we have: $$3y - z = 7 \quad \text{(Equation 4)}$$ $$x + 5y = 11 \quad \text{(Equation 5)}$$ From Equation 4: $$z = 3y - 7$$ Substitute this into Equation 1: $$x + y - 2(3y - 7) = 5$$ Simplify and solve for $$x$$: $$x + y - 6y + 14 = 5$$ $$x - 5y = -9 \quad \text{(Equation 6)}$$ ### Step 4: Solve for $$x$$ and $$y$$ Now, solve the system formed by Equation 5 and Equation 6: $$x + 5y = 11 \quad \text{(Equation 5)}$$ $$x - 5y = -9 \quad \text{(Equation 6)}$$ Add these two equations: $$2x = 2 \quad \Rightarrow \quad x = 1$$ Substitute $$x = 1$$ back into Equation 5: $$1 + 5y = 11 \quad \Rightarrow \quad 5y = 10 \quad \Rightarrow \quad y = 2$$ ### Step 5: Solve for $$z$$ Substitute $$y = 2$$ into the expression for $$z$$ from Equation 4: $$z = 3(2) - 7 = 6 - 7 = -1$$ ### Final Solution The solution to the system is: $$x = 1, \quad y = 2, \quad z = -1$$ Would you like more details, or do you have any questions? --- Here are 5 related questions: 1. How can you verify the solution of a system of linear equations? 2. What are the methods to solve a system of three linear equations? 3. How do you represent a system of linear equations using matrices? 4. What is the determinant of a matrix, and how is it useful in solving systems of equations? 5. What are the conditions for a system of linear equations to have a unique solution? **Tip:** Always substitute your solution back into the original equations to verify correctness.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Matrix Operations
Formulas
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Theorems
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Suitable Grade Level
High School
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