Math Problem Statement
Solve the system of equations: x - 2y + 2z = 9, -x + 3y = -4, 2x - 5y + 3z = 16.
Solution
The system of equations provided is:
1. \quad & x - 2y + 2z = 9 \\ 2. \quad & -x + 3y = -4 \\ 3. \quad & 2x - 5y + 3z = 16 \end{aligned}$$ ### Step 1: Solve equation (2) for $$x$$ From equation (2): $$-x + 3y = -4 \implies x = 3y + 4$$ ### Step 2: Substitute $$x = 3y + 4$$ into equations (1) and (3) #### Substituting into equation (1): $$(3y + 4) - 2y + 2z = 9$$ Simplifying: $$3y - 2y + 4 + 2z = 9 \implies y + 2z = 5$$ This is a new equation: $$(4) \quad y + 2z = 5$$ #### Substituting into equation (3): $$2(3y + 4) - 5y + 3z = 16$$ Simplifying: $$6y + 8 - 5y + 3z = 16 \implies y + 3z = 8$$ This is a new equation: $$(5) \quad y + 3z = 8$$ ### Step 3: Solve the system of equations (4) and (5) Subtract equation (4) from equation (5): $$(y + 3z) - (y + 2z) = 8 - 5 \implies z = 3$$ ### Step 4: Substitute $$z = 3$$ into equation (4) to find $$y$$ $$y + 2(3) = 5 \implies y + 6 = 5 \implies y = -1$$ ### Step 5: Substitute $$y = -1$$ into $$x = 3y + 4$$ to find $$x$$ $$x = 3(-1) + 4 = -3 + 4 = 1$$ ### Final Solution: The solution to the system is: $$(x, y, z) = (1, -1, 3)$$ Let me know if you need further clarification or details! Here are some related questions for practice: 1. What is the general approach for solving systems of equations using substitution? 2. How would you solve the system using the elimination method? 3. Can you verify the solution by substituting $$x = 1$$, $$y = -1$$, and $$z = 3$$ into the original equations? 4. How do systems of equations differ when they have no solution or infinitely many solutions? 5. What are the advantages of using substitution versus elimination? **Tip**: Always double-check your solution by substituting it back into all original equations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Equations
Algebra
Substitution Method
Formulas
Substitution method: solve one equation for one variable, substitute into others
Theorems
Linear systems of equations theory
Suitable Grade Level
Grades 10-12
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