Math Problem Statement

Реши систему уравнений методом Гаусса

Реши систему уравнений методом Гаусса

Solution

We are given the following system of linear equations:

x + 5z = 0 \\ 3x - 2y + 7z = 4 \\ 5x + 9z = 16 \end{cases}$$ ### Step 1: Write the system as an augmented matrix The augmented matrix for this system is: $$\begin{pmatrix} 1 & 0 & 5 & | & 0 \\ 3 & -2 & 7 & | & 4 \\ 5 & 0 & 9 & | & 16 \end{pmatrix}$$ ### Step 2: Apply Gaussian elimination #### Step 2.1: Make the first pivot (top-left) element a 1 (already done). #### Step 2.2: Eliminate the first column for rows 2 and 3. To eliminate the $$x$$-terms in rows 2 and 3, we perform the following row operations: - $$R_2 \leftarrow R_2 - 3R_1$$ - $$R_3 \leftarrow R_3 - 5R_1$$ This gives the matrix: $$\begin{pmatrix} 1 & 0 & 5 & | & 0 \\ 0 & -2 & -8 & | & 4 \\ 0 & 0 & -16 & | & 16 \end{pmatrix}$$ #### Step 2.3: Eliminate the second column for row 3. We will now eliminate the $$y$$-term in row 3 using the pivot in row 2: - $$R_3 \leftarrow R_3 - \frac{R_2}{2}$$ This gives the matrix: $$\begin{pmatrix} 1 & 0 & 5 & | & 0 \\ 0 & -2 & -8 & | & 4 \\ 0 & 0 & 0 & | & 0 \end{pmatrix}$$ ### Step 3: Back substitution Now, we solve the system using back substitution. 1. From the third row, we see that $$0 = 0$$, which implies that the system has infinitely many solutions. 2. From the second row: $$-2y - 8z = 4 \quad \Rightarrow \quad y = -2 - 4z$$ 3. From the first row: $$x + 5z = 0 \quad \Rightarrow \quad x = -5z$$ ### Step 4: General solution Thus, the general solution of the system is: $$x = -5z, \quad y = -2 - 4z, \quad z = z$$ This is a parametric solution where $$z$$ is a free parameter. Would you like more details on Gaussian elimination, or do you have any questions? Here are 5 related questions: 1. How can we represent the parametric solution geometrically? 2. How would the solution change if we added a constraint on $$z$$? 3. Can Gaussian elimination always be applied to any system of linear equations? 4. How does Gaussian elimination handle inconsistent systems? 5. What is the difference between Gaussian elimination and Gauss-Jordan elimination? **Tip**: Always check for zero rows when solving systems using Gaussian elimination, as they can indicate infinitely many solutions or no solution.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Gaussian Elimination
Systems of Equations

Formulas

Augmented Matrix
Row Operations for Gaussian Elimination

Theorems

Gaussian Elimination
Parametric Solution of Linear Systems

Suitable Grade Level

Grades 10-12

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