Math Problem Statement
Solution
The problem asks us to test for the consistency of the given system of equations and, if consistent, solve it. Let's go step by step:
Given System of Equations:
Testing for Consistency:
To test for consistency, we use the augmented matrix of the system of equations. The goal is to check if the system has solutions (consistent) or no solutions (inconsistent).
Step 1: Write the augmented matrix.
2 & -3 & 5 & 1 \\ 3 & 1 & -1 & 2 \\ 1 & 4 & -6 & 1 \end{bmatrix}$$ #### Step 2: Perform Gaussian elimination. 1. **Normalize Row 1 (R1):** Divide Row 1 by $$2$$: $$R1 \to \frac{R1}{2}: \begin{bmatrix} 1 & -3/2 & 5/2 & 1/2 \\ 3 & 1 & -1 & 2 \\ 1 & 4 & -6 & 1 \end{bmatrix}.$$ 2. **Eliminate the first variable (x) from R2 and R3:** $$R2 \to R2 - 3R1,\quad R3 \to R3 - R1$$ After performing these row operations, the matrix becomes: $$\begin{bmatrix} 1 & -3/2 & 5/2 & 1/2 \\ 0 & 11/2 & -19/2 & 1/2 \\ 0 & 11/2 & -17/2 & 1/2 \end{bmatrix}.$$ 3. **Eliminate the second variable (y) from R3:** Subtract $$R2$$ from $$R3$$: $$R3 \to R3 - R2.$$ This results in: $$\begin{bmatrix} 1 & -3/2 & 5/2 & 1/2 \\ 0 & 11/2 & -19/2 & 1/2 \\ 0 & 0 & 1 & 0 \end{bmatrix}.$$ --- ### Solution of the System: From the matrix, the equations can now be rewritten as: 1. $$z = t$$ (free parameter) 2. $$\frac{11}{2}y - \frac{19}{2}z = \frac{1}{2}$$ 3. $$x - \frac{3}{2}y + \frac{5}{2}z = \frac{1}{2}$$ #### Solving for $$y$$ and $$x$$: 1. From the second equation: $$11y - 19z = 1 \implies y = \frac{1 + 19z}{11}.$$ 2. From the first equation: $$x - \frac{3}{2}y + \frac{5}{2}z = \frac{1}{2}.$$ Substituting $$y = \frac{1 + 19z}{11}$$: $$x = \frac{7}{11} - \frac{2z}{11}.$$ --- ### Final Parametric Solution: The solution is: $$x = \frac{7}{11} - \frac{2t}{11},\quad y = \frac{1}{11} + \frac{17t}{11},\quad z = t,$$ where $$t$$ is a free parameter. The system is **consistent**. --- ### Do you want further clarification or details? Let me know! Here are 5 related questions for exploration: 1. What is the difference between consistent and inconsistent systems of equations? 2. How is Gaussian elimination used to solve linear systems? 3. What does it mean when a system has infinitely many solutions? 4. How can we verify the solution of this system of equations? 5. How can parametric solutions be visualized in three dimensions? **Tip:** When solving systems with parameters, always interpret the solution geometrically to understand the nature of the solutions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Linear Equations
Matrix Algebra
Gaussian Elimination
Formulas
Gaussian elimination steps for solving linear systems
Parametric representation of solutions
Theorems
Consistency of linear systems
Row reduction in matrices
Suitable Grade Level
Undergraduate (Linear Algebra Level)
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