Math Problem Statement
Solution
Let's solve the system of equations using augmented matrix methods. The system provided is:
Step 1: Set up the augmented matrix
The system of equations can be represented as an augmented matrix:
2 & -1 & | & 7 \\ 1 & 5 & | & 20 \end{pmatrix}$$ ### Step 2: Perform row operations to get the matrix into row-echelon form 1. Leave the first row as it is: $$R_1: \quad 2 \quad -1 \quad | \quad 7$$ 2. Use row operations to eliminate $$x_1$$ from the second row. Multiply $$R_2$$ by 2 and subtract $$R_1$$: $$R_2 \rightarrow 2R_2 - R_1 = (2 \cdot 1, 2 \cdot 5, 2 \cdot 20) - (2, -1, 7)$$ This gives: $$0 \quad 11 \quad | \quad 33$$ Now the augmented matrix is: $$\begin{pmatrix} 2 & -1 & | & 7 \\ 0 & 11 & | & 33 \end{pmatrix}$$ ### Step 3: Solve for the variables From the second row, we can solve for $$x_2$$: $$11x_2 = 33 \quad \Rightarrow \quad x_2 = \frac{33}{11} = 3$$ Substitute $$x_2 = 3$$ into the first row equation: $$2x_1 - 3 = 7 \quad \Rightarrow \quad 2x_1 = 10 \quad \Rightarrow \quad x_1 = \frac{10}{2} = 5$$ ### Final solution: The unique solution is: $$x_1 = 5, \quad x_2 = 3$$ Thus, the correct answer is **A**: The unique solution is $$x_1 = 5$$ and $$x_2 = 3$$. --- Do you want further details or have any questions? Here are 5 related questions: 1. How do we know when a system has no solutions using augmented matrices? 2. How can you tell if a system has infinitely many solutions using row reduction? 3. What are the steps to convert a system to row-reduced echelon form? 4. How can augmented matrices be used for larger systems (3x3 or more)? 5. What do inconsistent rows indicate in terms of the system of equations? **Tip**: Always check the determinant of the coefficient matrix to see if the system is consistent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
System of Equations
Augmented Matrices
Row-Echelon Form
Formulas
Augmented Matrix Setup
Row Reduction
Back Substitution
Theorems
Row Reduction Theorem
Existence and Uniqueness Theorem
Suitable Grade Level
Grade 10-12 or College Level
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