Math Problem Statement
Решить систему методом Гаусса:
\[
\begin{aligned}
-5x_1 + 5x_2 - 2x_3 - x_4 &= 3, \\
x_1 - 4x_2 + x_3 + 2x_4 &= 3, \\
2x_1 - 5x_2 + 2x_3 &= 1.
\end{aligned}
\]
Solution
The problem involves solving a system of linear equations using Gaussian elimination. The system is:
-5x_1 + 5x_2 - 2x_3 - x_4 &= 3, \\ x_1 - 4x_2 + x_3 + 2x_4 &= 3, \\ 2x_1 - 5x_2 + 2x_3 &= 1. \end{aligned}$$ ### Step-by-Step Solution Using Gaussian Elimination 1. **Write the augmented matrix:** $$\begin{bmatrix} -5 & 5 & -2 & -1 & 3 \\ 1 & -4 & 1 & 2 & 3 \\ 2 & -5 & 2 & 0 & 1 \end{bmatrix}.$$ 2. **Perform row operations to form an upper triangular matrix.** Start by making the first pivot $$a_{11} = 1$$: Divide Row 2 by $$1$$ (no change) and scale Row 1 by $$-1/5$$. 3. **Eliminate below the pivot in the first column** to set elements $$a_{21}$$ and \(a_{31}\Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Systems of Linear Equations
Formulas
Row operations to create an upper triangular matrix
Back-substitution to solve for variables
Theorems
Gaussian elimination method
Suitable Grade Level
Undergraduate Level (Linear Algebra)