Math Problem Statement
Solve using augmented matrices.
2
2
x 1
x1
plus
3 x 2 equals
3x2=
11
11
x 1
x1
minus
−
x 2
x2
equals
=
negative 2
−2
A. The unique solution is x1=enter your response here and x2=enter your response here.
B. The system has infinitely many solutions. The solution is x1=enter your response here and x2=t.
(Simplify your answer. Type an expression using t as the variable.)
C. There is no solution.
Solution
We are given the system of linear equations:
1. \quad 2x_1 + 3x_2 &= 11 \\ 2. \quad x_1 - x_2 &= -2 \end{aligned}$$ We can solve this system using augmented matrices. ### Step 1: Write the augmented matrix The augmented matrix for this system is: $$\begin{pmatrix} 2 & 3 & | & 11 \\ 1 & -1 & | & -2 \end{pmatrix}$$ ### Step 2: Perform row operations to reduce the matrix **Operation 1:** Subtract $$\frac{1}{2}$$ of row 1 from row 2 to eliminate $$x_1$$ in the second row. $$R_2 \to R_2 - \frac{1}{2} R_1$$ $$\begin{pmatrix} 2 & 3 & | & 11 \\ 1 & -1 & | & -2 \end{pmatrix} \quad \Rightarrow \quad \begin{pmatrix} 2 & 3 & | & 11 \\ 0 & -\frac{7}{2} & | & -\frac{15}{2} \end{pmatrix}$$ **Operation 2:** Multiply row 2 by $$-\frac{2}{7}$$ to simplify the second row. $$R_2 \to -\frac{2}{7} R_2$$ $$\begin{pmatrix} 2 & 3 & | & 11 \\ 0 & 1 & | & \frac{15}{7} \end{pmatrix}$$ **Operation 3:** Subtract $$3$$ times row 2 from row 1 to eliminate $$x_2$$ in the first row. $$R_1 \to R_1 - 3 R_2$$ $$\begin{pmatrix} 2 & 0 & | & \frac{8}{7} \\ 0 & 1 & | & \frac{15}{7} \end{pmatrix}$$ ### Step 3: Solve for $$x_1$$ and $$x_2$$ From the matrix, we have: $$2x_1 = \frac{8}{7} \quad \Rightarrow \quad x_1 = \frac{4}{7}$$ $$x_2 = \frac{15}{7}$$ Thus, the unique solution is: $$x_1 = \frac{4}{7}, \quad x_2 = \frac{15}{7}$$ ### Answer: **A.** The unique solution is $$x_1 = \frac{4}{7}$$ and $$x_2 = \frac{15}{7}$$. --- Would you like more details on any step? Here are 5 related questions you might explore next: 1. How do you solve systems of equations using Gaussian elimination? 2. What are the conditions for a system to have infinitely many solutions? 3. How do you represent a system of equations as an augmented matrix? 4. Can augmented matrices be used for non-linear systems of equations? 5. What is the difference between row echelon form and reduced row echelon form? **Tip:** Always check your row operations carefully when solving with matrices to avoid arithmetic errors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Augmented Matrices
Row Reduction
Formulas
Row operations on augmented matrices
Solving linear systems using matrix reduction
Theorems
Gaussian Elimination
Unique Solution Criterion
Suitable Grade Level
Grades 10-12
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