Math Problem Statement
Solve the system of linear equations depending on the parameter \( \lambda \):
\begin{cases} 2x_1 - x_2 + 3x_3 + 4x_4 = 5, \\ 4x_1 - 2x_2 + 5x_3 + 6x_4 = 7, \\ 6x_1 - 3x_2 + 7x_3 + 8x_4 = 9, \\ \lambda x_1 - 4x_2 + 9x_3 + 10x_4 = 11. \end{cases}
Solution
The given system of linear equations is:
2x_1 - x_2 + 3x_3 + 4x_4 = 5, \\ 4x_1 - 2x_2 + 5x_3 + 6x_4 = 7, \\ 6x_1 - 3x_2 + 7x_3 + 8x_4 = 9, \\ \lambda x_1 - 4x_2 + 9x_3 + 10x_4 = 11. \end{cases}$$ The task is to find the general solution of this system, which depends on the parameter $$\lambda$$. ### Approach: 1. **Set up the augmented matrix** corresponding to the system. 2. **Row-reduce** the matrix (using Gaussian elimination) to bring it to row echelon or reduced row echelon form. 3. Analyze how the parameter $$\lambda$$ affects the solution (e.g., if certain values of $$\lambda$$ result in no solutions or infinite solutions). 4. **Back-substitute** (if needed) to express the general solution in terms of free variables or parameters. Let me first convert the system into an augmented matrix and perform the row-reduction steps to find the general solution. I'll work through this now.After performing Gaussian elimination, the system's augmented matrix in row-reduced echelon form (RREF) is: $$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 & 4 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$$ This corresponds to the following system of equations: $$\begin{cases} x_1 = 0, \\ x_2 + 2x_4 = 4, \\ x_3 + 2x_4 = 3. \end{cases}$$ Thus, the free variable here is $$x_4$$. Solving for $$x_2$$ and $$x_3$$ in terms of $$x_4$$, we get: $$x_2 = 4 - 2x_4, \quad x_3 = 3 - 2x_4, \quad x_1 = 0.$$ ### General solution: $$x_1 = 0, \quad x_2 = 4 - 2x_4, \quad x_3 = 3 - 2x_4, \quad x_4 = x_4.$$ This is the general solution for the system, which depends on the free variable $$x_4$$. The parameter $$\lambda$$ does not restrict the solution as it does not appear in a singular manner after row reduction. Would you like any further details or clarifications on this solution? --- Here are 5 related questions to expand your understanding: 1. How does the presence of a free variable like $$x_4$$ affect the nature of the solution? 2. Can there be cases where the parameter $$\lambda$$ leads to no solution? Why or why not? 3. What does it mean for a system to be consistent or inconsistent, and how do we check for consistency in this system? 4. How would the general solution change if $$\lambda$$ had a specific value, such as $$\lambda = 0$$? 5. How can the solution be interpreted geometrically in terms of the dimensionality of the solution space? **Tip:** When solving a system of equations, always check for special cases (like singular matrices) that might affect the existence or uniqueness of solutions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Formulas
Gaussian elimination process
Row reduction
Theorems
Rank Theorem
System Consistency Theorem
Suitable Grade Level
University level, advanced linear algebra
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