Math Problem Statement
Using gauss elimination method
Solve the following system of linear equations using the Gaussian elimination method:
1. x1 + 2x2 + 3x3 + 4x4 = 32
2. 2x1 - x2 + 2x3 - x4 = 3
Solution
We are given a system of linear equations:
1. &\quad x_1 + 2x_2 + 3x_3 + 4x_4 = 32, \\ 2. &\quad 2x_1 - x_2 + 2x_3 - x_4 = 3. \end{aligned}$$ To solve this using **Gaussian elimination**, we will follow these steps: ### Step 1: Set up the augmented matrix We can represent the system of equations as an augmented matrix: $$\begin{pmatrix} 1 & 2 & 3 & 4 & | & 32 \\ 2 & -1 & 2 & -1 & | & 3 \end{pmatrix}$$ ### Step 2: Perform row operations to reduce the matrix to upper triangular form We aim to make the matrix upper triangular by eliminating the elements below the pivot in the first column. - Start by subtracting 2 times the first row from the second row to eliminate the $$2x_1$$ term in the second equation. $$R_2 \to R_2 - 2R_1$$ The augmented matrix becomes: $$\begin{pmatrix} 1 & 2 & 3 & 4 & | & 32 \\ 0 & -5 & -4 & -9 & | & -61 \end{pmatrix}$$ ### Step 3: Back substitution Now we have an upper triangular matrix, and we can solve by back substitution. 1. From the second row: $$-5x_2 - 4x_3 - 9x_4 = -61$$ This gives the equation: $$x_2 = \frac{-61 + 4x_3 + 9x_4}{-5}$$ 2. From the first row: $$x_1 + 2x_2 + 3x_3 + 4x_4 = 32$$ Substituting $$x_2$$ into this equation, we can solve for $$x_1$$. This system is underdetermined (fewer equations than variables), so it will have a family of solutions where $$x_3$$ and $$x_4$$ can take on free values. From here, you can assign arbitrary values to $$x_3$$ and $$x_4$$, then compute the corresponding values of $$x_1$$ and $$x_2$$. Would you like me to solve for specific values of $$x_3$$ and $$x_4$$ or explain further details? Here are five follow-up questions: 1. Would you like to specify values for $$x_3$$ and $$x_4$$ to compute a particular solution? 2. Do you need help interpreting the meaning of free variables in the solution? 3. Shall weAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Systems of Linear Equations
Formulas
Augmented matrix
Row operations for elimination
Theorems
Gaussian elimination
Suitable Grade Level
Undergraduate or advanced high school (Grades 11-12)
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