Math Problem Statement
x-y+z=-1;x+y+3z=-3;2x-y+2z=0
Solution
We are given the following system of linear equations:
1. &\quad x - y + z = -1 \\ 2. &\quad x + y + 3z = -3 \\ 3. &\quad 2x - y + 2z = 0 \end{aligned}$$ We will solve this system using either substitution or elimination. Let's proceed step-by-step with the elimination method. ### Step 1: Eliminate $$y$$ from the first two equations Add equations (1) and (2): $$(x - y + z) + (x + y + 3z) = (-1) + (-3)$$ Simplifying: $$2x + 4z = -4$$ This gives us a new equation: $$4. \quad 2x + 4z = -4 \quad \text{or} \quad x + 2z = -2$$ ### Step 2: Eliminate $$y$$ from the first and third equations Now, subtract equation (1) from equation (3): $$(2x - y + 2z) - (x - y + z) = 0 - (-1)$$ Simplifying: $$x + z = 1$$ This gives us another new equation: $$5. \quad x + z = 1$$ ### Step 3: Solve the system of two equations Now we have the system: $$\begin{aligned} 4. &\quad x + 2z = -2 \\ 5. &\quad x + z = 1 \end{aligned}$$ Subtract equation (5) from equation (4): $$(x + 2z) - (x + z) = -2 - 1$$ Simplifying: $$z = -3$$ ### Step 4: Substitute $$z = -3$$ back into equation (5) Substitute into $$x + z = 1$$: $$x - 3 = 1$$ Solving for $$x$$: $$x = 4$$ ### Step 5: Substitute $$x = 4$$ and $$z = -3$$ into equation (1) Substitute into $$x - y + z = -1$$: $$4 - y - 3 = -1$$ Simplifying: $$1 - y = -1$$ Solving for $$y$$: $$y = 2$$ ### Final Solution The solution to the system is: $$x = 4, \quad y = 2, \quad z = -3$$ Would you like further details on any of the steps? --- Here are 5 related questions you can try: 1. Solve the system $$x + 2y - z = 1$$, $$2x - y + z = 3$$, $$x - y - z = -1$$. 2. How can the substitution method be used to solve this system of equations? 3. What are the benefits of using the elimination method in solving systems of equations? 4. What would happen if one of the equations in a system is a multiple of another? 5. Can you solve a system of equations graphically, and how would it look for this case? **Tip**: For systems of linear equations, checking the consistency of your solution (by substituting back) is a good habit to ensure accuracy.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Elimination Method
Substitution Method
Formulas
x - y + z = -1
x + y + 3z = -3
2x - y + 2z = 0
Equation addition and subtraction for elimination: (x - y + z) + (x + y + 3z)
Substitution of variables: x + z = 1
Theorems
Linear Equations Theorem
Consistency of Linear Systems
Suitable Grade Level
Grades 9-11
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