Math Problem Statement
Solve the system of linear equations and determine if there is no solution, one solution, or infinitely many solutions:
x_1 + x_2 + x_3 = 1
x_1 + 2x_2 + 2x_3 = 3
x_1 + x_2 + x_3 = 4
Solution
The problem in the image asks you to find a complete solution to the system of linear equations, and then determine whether there is no solution, one solution, or infinitely many solutions. The system of equations is:
x_1 + x_2 + x_3 & = 1 \\ x_1 + 2x_2 + 2x_3 & = 3 \\ x_1 + x_2 + x_3 & = 4 \end{align*}$$ ### Step 1: Write the system in matrix form The system can be represented as an augmented matrix: $$\begin{pmatrix} 1 & 1 & 1 & | & 1 \\ 1 & 2 & 2 & | & 3 \\ 1 & 1 & 1 & | & 4 \end{pmatrix}$$ ### Step 2: Perform row operations to simplify the matrix We aim to row-reduce the matrix to its simplest form (Row Echelon Form). 1. Subtract the first row from the second row: $$R2 = R2 - R1 \implies \begin{pmatrix} 1 & 1 & 1 & | & 1 \\ 0 & 1 & 1 & | & 2 \\ 1 & 1 & 1 & | & 4 \end{pmatrix}$$ 2. Subtract the first row from the third row: $$R3 = R3 - R1 \implies \begin{pmatrix} 1 & 1 & 1 & | & 1 \\ 0 & 1 & 1 & | & 2 \\ 0 & 0 & 0 & | & 3 \end{pmatrix}$$ ### Step 3: Analyze the row-reduced matrix The row-reduced matrix is: $$\begin{pmatrix} 1 & 1 & 1 & | & 1 \\ 0 & 1 & 1 & | & 2 \\ 0 & 0 & 0 & | & 3 \end{pmatrix}$$ The third row translates to the equation $$0x_1 + 0x_2 + 0x_3 = 3$$, which simplifies to $$0 = 3$$. This is a contradiction, which indicates that there is **no solution** to this system of equations. ### Conclusion Given that the system leads to a contradiction, the correct answer is: **(C) There is no solution. The system is inconsistent.** --- Would you like more details on any of the steps? Here are 5 additional questions you might explore: 1. What does it mean for a system of equations to be inconsistent? 2. How do you interpret an augmented matrix in the context of linear equations? 3. What are the steps to converting a system of linear equations to Row Echelon Form? 4. What would indicate a system has infinitely many solutions? 5. Can you provide an example of a system that has a unique solution? **Tip:** When solving systems of equations, always check for consistency by fully reducing the matrix and interpreting the results correctly.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Row Echelon Form
Formulas
Matrix row operations
Theorems
No solution theorem for inconsistent systems
Suitable Grade Level
Grades 10-12
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