Math Problem Statement
Solution
Let's break down the problem from the image you uploaded:
Problem Breakdown:
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System of Equations Given:
2x - y - 2z &= 4, \\ 3x + y + z &= 5, \\ 4x + 2y - z &= 6. \end{aligned}$$ -
Create the Augmented Matrix: The augmented matrix for the system of equations is:
2 & -1 & -2 & \vert & 4 \\ 3 & 1 & 1 & \vert & 5 \\ 4 & 2 & -1 & \vert & 6 \end{bmatrix}$$ -
Convert to Row-Echelon Form (REF): The row-echelon form (REF) involves using Gaussian elimination. The goal is to get a matrix where the left part forms an upper triangular matrix.
After performing the necessary row operations:
2 & -1 & -2 & \vert & 4 \\ 0 & 5 & 5 & \vert & -1 \\ 0 & 0 & -10 & \vert & -10 \end{bmatrix}$$ -
Interpret the Results: The next steps involve identifying whether the system is consistent or inconsistent, dependent or independent.
Corresponding Simplified System:
The simplified matrix corresponds to the system:
2x - y - 2z &= 4, \\ 5y + 5z &= -1, \\ -10z &= -10. \end{aligned}$$ ### Solving the System: - From the third equation: $$z = 1$$. - Substitute $$z = 1$$ into the second equation to find $$y$$: $$5y + 5(1) = -1 \Rightarrow 5y = -6 \Rightarrow y = -\frac{6}{5}.$$ - Substitute $$y = -\frac{6}{5}$$ and $$z = 1$$ into the first equation to find $$x$$: $$2x - \left(-\frac{6}{5}\right) - 2(1) = 4 \Rightarrow 2x + \frac{6}{5} - 2 = 4.$$ Simplifying: $$2x = 4 - \frac{6}{5} + 2 = \frac{18}{5} \Rightarrow x = \frac{9}{5}.$$ ### Final Answer: The solution to the system is $$x = \frac{9}{5}$$, $$y = -\frac{6}{5}$$, and $$z = 1$$. --- Would you like more detailed steps for the row operations, or do you have any other questions? Here are five related questions to deepen your understanding: 1. How do you determine if a system of equations is consistent or inconsistent? 2. What are the conditions under which a system is dependent or independent? 3. Can you explain how to perform row operations to achieve the row-echelon form? 4. What is the difference between row-echelon form and reduced row-echelon form? 5. How do you interpret a solution set when a system has infinitely many solutions? **Tip:** When performing row operations, it's essential to keep track of the operations to avoid mistakes and easily backtrack if needed.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Row-Echelon Form
Gaussian Elimination
Formulas
Augmented Matrix
Row Operations
Back-Substitution
Theorems
Gaussian Elimination
Row-Echelon Form
Suitable Grade Level
Grades 10-12