Math Problem Statement
answer in latex
Solve the system: { x1 + x2 - 2x3 = -1, 3x1 + 2x2 - 2x3 = 2 }
Solution
To solve the given system of linear equations, we will proceed by writing the system in matrix form and then performing row operations.
The system is:
x_1 + x_2 - 2x_3 = -1, \\ 3x_1 + 2x_2 - 2x_3 = 2. \end{cases}$$ ### Step 1: Convert the System to an Augmented Matrix The augmented matrix for this system is: $$\begin{bmatrix} 1 & 1 & -2 & | & -1 \\ 3 & 2 & -2 & | & 2 \\ \end{bmatrix}$$ ### Step 2: Perform Row Operations 1. Subtract 3 times the first row from the second row to eliminate $$x_1$$ from the second equation: $$R_2 = R_2 - 3 \times R_1$$ This gives: $$\begin{bmatrix} 1 & 1 & -2 & | & -1 \\ 0 & -1 & 4 & | & 5 \\ \end{bmatrix}$$ 2. Now, divide the second row by $$-1$$ to simplify: $$R_2 = -1 \times R_2$$ This gives: $$\begin{bmatrix} 1 & 1 & -2 & | & -1 \\ 0 & 1 & -4 & | & -5 \\ \end{bmatrix}$$ 3. Finally, subtract the second row from the first row to eliminate $$x_2$$ from the first equation: $$R_1 = R_1 - R_2$$ This results in: $$\begin{bmatrix} 1 & 0 & 2 & | & 4 \\ 0 & 1 & -4 & | & -5 \\ \end{bmatrix}$$ ### Step 3: Rewrite the System Now, the system of equations is: $$\begin{cases} x_1 + 2x_3 = 4, \\ x_2 - 4x_3 = -5. \end{cases}$$ ### Step 4: Express in Terms of $$t$$ (Free Variable) Let $$x_3 = t$$, where $$t$$ is a free variable. Then: 1. From the first equation: $$x_1 = 4 - 2t$$. 2. From the second equation: $$x_2 = -5 + 4t$$. ### Solution The solution in vector form is: $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 4 \\ -5 \\ 0 \end{bmatrix} + t \begin{bmatrix} -2 \\ 4 \\ 1 \end{bmatrix},$$ where $$t$$ is a free variable. This is the complete solution to the system. --- Would you like further details or have any questions? ### Related Questions: 1. How do you identify free variables in a system of linear equations? 2. What are the general steps to solve a system of equations by the matrix method? 3. How does row reduction help in finding solutions for systems with free variables? 4. What does it mean if a system has infinitely many solutions? 5. How can you verify the solution for a system of linear equations? ### Tip: When solving systems with free variables, remember that each free variable corresponds to a degree of freedom in the solution set, leading to infinitely many solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Row Reduction
Formulas
Row reduction
Matrix operations
Theorems
Gaussian Elimination
Suitable Grade Level
Grades 10-12
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