Math Problem Statement
Identify the number of solutions of the system of linear equations and solve it using elimination.
Solution
The system of linear equations given in the image is:
1. \quad x + 3y - z &= 2 \\ 2. \quad x + y - z &= 0 \\ 3. \quad 3x + 2y - 3z &= -1 \end{aligned}$$ ### Step 1: Use the Elimination Method We aim to eliminate variables by manipulating the equations. #### Subtract Equation (2) from Equation (1): $$(x + 3y - z) - (x + y - z) = 2 - 0$$ Simplifying this: $$(3y - y) = 2 \quad \Rightarrow \quad 2y = 2 \quad \Rightarrow \quad y = 1$$ Now that we have $$y = 1$$, substitute this value back into both Equation (1) and Equation (2) to solve for $$x$$ and $$z$$. #### Step 2: Substitute $$y = 1$$ into Equation (2): $$x + 1 - z = 0 \quad \Rightarrow \quad x - z = -1 \quad \Rightarrow \quad x = z - 1$$ #### Step 3: Substitute $$y = 1$$ into Equation (1): $$x + 3(1) - z = 2 \quad \Rightarrow \quad x + 3 - z = 2 \quad \Rightarrow \quad x - z = -1$$ This confirms $$x = z - 1$$ again, so the solution is consistent. #### Step 4: Substitute $$x = z - 1$$ and $$y = 1$$ into Equation (3): $$3(z - 1) + 2(1) - 3z = -1$$ Simplifying: $$3z - 3 + 2 - 3z = -1 \quad \Rightarrow \quad -1 = -1$$ This is a true statement, so the system is consistent and dependent, meaning there are **infinitely many solutions**. ### Final Solution: The general solution in terms of $$z$$ is: $$(x, y, z) = (z - 1, 1, z)$$ --- Would you like more details or have any questions? Here are 5 related questions: 1. How do you know if a system of equations has no solution? 2. Can the elimination method be used for any system of equations? 3. What are the conditions for a system to have infinitely many solutions? 4. How would you solve the same system using substitution instead of elimination? 5. How do the equations in a system change geometrically when they lead to no solution? **Tip**: Always check for consistency in the equations after eliminating variables to confirm if a system has infinitely many solutions or exactly one.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Linear Equations
Elimination Method
Infinitely Many Solutions
Formulas
Elimination method for solving systems of equations
Theorems
Consistency and dependency of linear equations
Suitable Grade Level
Grades 9-12
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