Math Problem Statement

реши пример

реши пример

Solution

The given image contains a system of three linear equations:

  1. x1−x2−x3+x4=3x_1 - x_2 - x_3 + x_4 = 3
  2. −3x1+x2−x3+2x4=−3-3x_1 + x_2 - x_3 + 2x_4 = -3
  3. 2x1−x2+x3−3x4=22x_1 - x_2 + x_3 - 3x_4 = 2

Solution:

To solve this system of equations, we can use the Gaussian elimination method or any matrix-based solver. Let’s solve step by step.

Step 1: Write the augmented matrix

1 & -1 & -1 & 1 & | & 3 \\ -3 & 1 & -1 & 2 & | & -3 \\ 2 & -1 & 1 & -3 & | & 2 \end{bmatrix}$$ #### Step 2: Solve using Gaussian elimination We will convert this matrix to row echelon form and back-substitute to find the solution. 1. Start with the first pivot (row 1, column 1): Divide row 1 by 1 (pivot remains unchanged): $$\begin{bmatrix} 1 & -1 & -1 & 1 & | & 3 \\ -3 & 1 & -1 & 2 & | & -3 \\ 2 & -1 & 1 & -3 & | & 2 \end{bmatrix}$$ 2. Eliminate column 1 for rows 2 and 3: - Row 2: Add $$3 \times \text{row 1}$$ to row 2. - Row 3: Subtract $$2 \times \text{row 1}$$ from row 3. Result: $$\begin{bmatrix} 1 & -1 & -1 & 1 & | & 3 \\ 0 & -2 & -4 & 5 & | & 6 \\ 0 & 1 & 3 & -5 & | & -4 \end{bmatrix}$$ 3. Use row 2 as the new pivot to eliminate below it: Divide row 2 by -2: \[ \begin{bmatrix} 1 & -1 & -1 &

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations

Formulas

Gaussian Elimination
Matrix Representation of Linear Systems

Theorems

Row Reduction for Solving Linear Systems

Suitable Grade Level

Grades 10-12

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