Math Problem Statement
−x + 2y − z = 0 −x − y + 2z = 0 2x − z = 3
Solution
You have a system of linear equations:
1. & \ -x + 2y - z = 0 \\ 2. & \ -x - y + 2z = 0 \\ 3. & \ 2x - z = 3 \end{aligned}$$ ### Solution Steps: #### Step 1: Solve equation 3 for $$z$$ From equation (3): $$2x - z = 3 \implies z = 2x - 3$$ #### Step 2: Substitute $$z = 2x - 3$$ into equations (1) and (2) - **Substitute into equation (1):** $$-x + 2y - (2x - 3) = 0$$ Simplifying: $$-x + 2y - 2x + 3 = 0 \implies -3x + 2y = -3 \implies 3x - 2y = 3 \quad \text{(Equation 4)}$$ - **Substitute into equation (2):** $$-x - y + 2(2x - 3) = 0$$ Simplifying: $$-x - y + 4x - 6 = 0 \implies 3x - y = 6 \quad \text{(Equation 5)}$$ #### Step 3: Solve equations (4) and (5) You now have: $$\begin{aligned} 4. & \ 3x - 2y = 3 \\ 5. & \ 3x - y = 6 \end{aligned}$$ Solve equation (5) for $$y$$: $$y = 3x - 6$$ #### Step 4: Substitute $$y = 3x - 6$$ into equation (4) Substitute into equation (4): $$3x - 2(3x - 6) = 3$$ Simplifying: $$3x - 6x + 12 = 3 \implies -3x = -9 \implies x = 3$$ #### Step 5: Find $$y$$ and $$z$$ - Substitute $$x = 3$$ into $$y = 3x - 6$$: $$y = 3(3) - 6 = 9 - 6 = 3$$ - Substitute $$x = 3$$ into $$z = 2x - 3$$: $$z = 2(3) - 3 = 6 - 3 = 3$$ ### Final Solution: $$x = 3, \quad y = 3, \quad z = 3$$ Would you like more details on any step? Here are 5 related questions you might find useful: 1. How do you solve systems of linear equations using matrices? 2. What is substitution method in solving systems of equations? 3. How does elimination method differ from substitution in solving equations? 4. Can a system of linear equations have no solution or infinite solutions? 5. What are the real-life applications of solving systems of equations? **Tip**: When solving systems of equations, always simplify expressions at every step to avoid errors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
System of Equations
Substitution Method
Formulas
z = 2x - 3
3x - 2y = 3
3x - y = 6
Theorems
Substitution Method in Solving Systems of Equations
Suitable Grade Level
Grades 9-11
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