Math Problem Statement
Решить систему линейных уравнений тремя способами: а) методом Гаусса;
Solution
I'll solve the given system of linear equations by using the Gaussian elimination method. Let's break the steps into a clear format.
System of Linear Equations
From the image, the system can be represented as:
4x - y + 2z &= 27 \\ 5x - y + 2z &= 33 \\ 2x + 3y + 2z &= 27 \end{aligned}$$ We'll proceed with Gaussian elimination. --- ### Step 1: Augmented Matrix We write the augmented matrix for the system: $$\begin{bmatrix} 4 & -1 & 2 & 27 \\ 5 & -1 & 2 & 33 \\ 2 & 3 & 2 & 27 \end{bmatrix}$$ --- ### Step 2: Row Reduction We'll perform row operations to bring the matrix to row-echelon form. #### 1. Make the first element of the first row (pivot) 1: Divide the first row by 4: $$\begin{bmatrix} 1 & -\frac{1}{4} & \frac{1}{2} & \frac{27}{4} \\ 5 & -1 & 2 & 33 \\ 2 & 3 & 2 & 27 \end{bmatrix}$$ #### 2. Eliminate the first element of rows 2 and 3: - $$R_2 \to R_2 - 5R_1$$ - $$R_3 \to R_3 - 2R_1$$ After these operations, the matrix becomes: $$\begin{bmatrix} 1 & -\frac{1}{4} & \frac{1}{2} & \frac{27}{4} \\ 0 & \frac{1}{4} & -\frac{1}{2} & \frac{3}{4} \\ 0 & \frac{13}{4} & 1 & \frac{81}{4} \end{bmatrix}$$ --- #### 3. Make the pivot in the second row a 1: Divide $$R_2$$ by $$\frac{1}{4}$$: $$\begin{bmatrix} 1 & -\frac{1}{4} & \frac{1}{2} & \frac{27}{4} \\ 0 & 1 & -2 & 3 \\ 0 & \frac{13}{4} & 1 & \frac{81}{4} \end{bmatrix}$$ #### 4. Eliminate the second element from the first and third rows: - $$R_1 \to R_1 + \frac{1}{4}R_2$$ - $$R_3 \to R_3 - \frac{13}{4}R_2$$ After these operations, the matrix becomes: $$\begin{bmatrix} 1 & 0 & \frac{3}{4} & \frac{30}{4} \\ 0 & 1 & -2 & 3 \\ 0 & 0 & 7 & 21 \end{bmatrix}$$ --- #### 5. Make the pivot in the third row a 1: Divide $$R_3$$ by 7: $$\begin{bmatrix} 1 & 0 & \frac{3}{4} & \frac{30}{4} \\ 0 & 1 & -2 & 3 \\ 0 & 0 & 1 & 3 \end{bmatrix}$$ #### 6. Eliminate the third element from the first and second rows: - $$R_1 \to R_1 - \frac{3}{4}R_3$$ - $$R_2 \to R_2 + 2R_3$$ The final row-reduced matrix is: $$\begin{bmatrix} 1 & 0 & 0 & 6 \\ 0 & 1 & 0 & 9 \\ 0 & 0 & 1 & 3 \end{bmatrix}$$ --- ### Solution From the matrix, we find: $$x = 6, \quad y = 9, \quad z = 3$$ Would you like me to verify the solution or solve using other methods, such as substitution or determinants? Let me know! Here are five related questions for further understanding: 1. How does Gaussian elimination differ from Gauss-Jordan elimination? 2. What are the benefits of using matrix methods to solve linear systems? 3. Can you solve the same system using Cramer's rule for practice? 4. How does pivoting improve numerical stability in Gaussian elimination? 5. What are some real-world applications of solving systems of linear equations? **Tip**: Always check the determinant of the coefficient matrix before solving; if it’s zero, the system might be inconsistent or dependent.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Gaussian Elimination
Formulas
Row operations for matrix reduction
Back substitution for solving equations
Theorems
Gaussian elimination process for solving linear systems
Suitable Grade Level
Grades 10-12, Undergraduate
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