Math Problem Statement
Solve the given system of linear equations and interpret the parametric solution.
Solution
The image shows a system of linear equations and its corresponding augmented matrix transformation. Let's analyze and solve it step by step.
System of Equations:
Augmented Matrix Representation:
The system can be represented as:
1 & 3 & 2 & 2 & | & 1 \\ 2 & 6 & 3 & 2 & | & 1 \end{bmatrix}$$ From the image, the row reduction step shows: $$\begin{bmatrix} 1 & 3 & 2 & 2 & | & 1 \\ 2 & 6 & 3 & 2 & | & 1 \end{bmatrix} \sim \begin{bmatrix} 1 & 3 & 2 & 2 & | & 1 \\ 0 & 0 & 1 & 2 & | & 1 \end{bmatrix}$$ ### Interpretation: The second row simplifies to: $$0x_1 + 0x_2 + x_3 + 2x_4 = 1 \quad \Rightarrow \quad x_3 + 2x_4 = 1.$$ The first row remains: $$x_1 + 3x_2 + 2x_3 + 2x_4 = 1.$$ ### Solution Process: 1. From the second equation ($$x_3 + 2x_4 = 1$$): $$x_3 = 1 - 2x_4.$$ 2. Substitute $$x_3 = 1 - 2x_4$$ into the first equation: $$x_1 + 3x_2 + 2(1 - 2x_4) + 2x_4 = 1.$$ Simplify: $$x_1 + 3x_2 + 2 - 4x_4 + 2x_4 = 1,$$ $$x_1 + 3x_2 - 2x_4 = -1.$$ 3. Rewrite the equations: - $$x_1 + 3x_2 - 2x_4 = -1$$ - $$x_3 = 1 - 2x_4$$ ### Parametric Solution: Let $$x_2 = t$$ and $$x_4 = s$$ (free variables). Then: $$x_1 = -1 - 3t + 2s,$$ $$x_3 = 1 - 2s,$$ $$x_2 = t,$$ $$x_4 = s.$$ ### Final Solution: $$x_1 = -1 - 3t + 2s, \quad x_2 = t, \quad x_3 = 1 - 2s, \quad x_4 = s,$$ where $$t, s \in \mathbb{R}$$. --- Would you like further clarification on any of these steps or a visualization? Here are five related questions for exploration: 1. How can we verify if the system has infinitely many solutions using the rank of the matrix? 2. What are the geometric interpretations of free variables $$t$$ and $$s$$? 3. How do row operations impact the solution space of a system of equations? 4. How can this parametric solution be written in vector form? 5. What are the implications if additional equations were added to this system? **Tip:** When solving systems of equations, always check the rank of the coefficient matrix and the augmented matrix to determine solution types (unique, none, or infinite).Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Row Reduction
Formulas
Row operations
Parametric form of solution for linear systems
Theorems
Solution classification using ranks of matrices
Suitable Grade Level
Grades 11-12 and Undergraduate
Related Recommendation
General Solution of a Linear System with Parameter t
Solving Systems of Equations from a Reduced Row-Echelon Matrix
General Solution of a System of Linear Equations in Parametric Vector Form
Solving System of Linear Equations Using Parametric Form
Solving a System of Linear Equations Using Row Reduction