Math Problem Statement
Solve the system of equestions
1. x1+x2+x3+x4= 0, 2x1+x2-x4= 0, x1+3x2+2x3+4x4= 0.
Solution
We are given the following system of equations:
Step 1: Write the system in matrix form
We can represent the system as:
1 & 1 & 1 & 1 \\ 2 & 1 & 0 & -1 \\ 1 & 3 & 2 & 4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$ ### Step 2: Solve using Gaussian elimination or row reduction We will perform row operations on the augmented matrix: $$\begin{bmatrix} 1 & 1 & 1 & 1 & 0 \\ 2 & 1 & 0 & -1 & 0 \\ 1 & 3 & 2 & 4 & 0 \end{bmatrix}$$ **(R2 - 2R1 → R2):** $$\begin{bmatrix} 1 & 1 & 1 & 1 & 0 \\ 0 & -1 & -2 & -3 & 0 \\ 1 & 3 & 2 & 4 & 0 \end{bmatrix}$$ **(R3 - R1 → R3):** $$\begin{bmatrix} 1 & 1 & 1 & 1 & 0 \\ 0 & -1 & -2 & -3 & 0 \\ 0 & 2 & 1 & 3 & 0 \end{bmatrix}$$ **(R2 → -R2):** $$\begin{bmatrix} 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 2 & 3 & 0 \\ 0 & 2 & 1 & 3 & 0 \end{bmatrix}$$ **(R3 - 2R2 → R3):** $$\begin{bmatrix} 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 2 & 3 & 0 \\ 0 & 0 & -3 & -3 & 0 \end{bmatrix}$$ ### Step 3: Back substitution From the last row, we have: $$-3x_3 - 3x_4 = 0 \quad \Rightarrow \quad x_3 + x_4 = 0 \quad \Rightarrow \quad x_3 = -x_4$$ Substitute $$x_3 = -x_4$$ into the second row: $$x_2 + 2x_3 + 3x_4 = 0 \quad \Rightarrow \quad x_2 + 2(-x_4) + 3x_4 = 0 \quad \Rightarrow \quad x_2 = -x_4$$ Substitute $$x_2 = -x_4$$ and $$x_3 = -x_4$$ into the first row: $$x_1 + x_2 + x_3 + x_4 = 0 \quad \Rightarrow \quad x_1 - x_4 - x_4 + x_4 = 0 \quad \Rightarrow \quad x_1 = x_4$$ ### Step 4: General solution The solution to the system is: $$x_1 = x_4, \quad x_2 = -x_4, \quad x_3 = -x_4, \quad x_4 = x_4$$ Thus, the general solution can be written as: $$(x_1, x_2, x_3, x_4) = (x_4, -x_4, -x_4, x_4) = x_4(1, -1, -1, 1)$$ where $$x_4$$ is a free parameter. ### Conclusion: The solution is a family of solutions of the form: $$(x_1, x_2, x_3, x_4) = t(1, -1, -1, 1), \quad t \in \mathbb{R}$$ Let me know if you need further details or have any questions! Here are some follow-up questions related to this: 1. What happens if we add a fourth equation to the system? 2. How can we visualize the solution in 4-dimensional space? 3. How does Gaussian elimination help in solving larger systems of equations? 4. How can we use matrices to express dependent or independent systems? 5. What does the free parameter $$t$$ represent geometrically? **Tip:** For underdetermined systems (fewer equations than variables), solutions often involve free parameters, indicating a family of solutions rather than a unique one.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Gaussian Elimination
Formulas
Gaussian elimination process for solving systems of linear equations
General form of a system of linear equations: Ax = b
Theorems
Row Reduction (Gaussian Elimination)
Free Variables in Linear Systems
Suitable Grade Level
Grades 10-12 (High School)
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