Solve the following system of equations: 2𝑥 + 3𝑦 + 4𝑧 = 11, 𝑥 + 5𝑦 + 7𝑧 = 15,
3𝑥 + 11𝑦 + 13𝑧 = 25 by Gaussian elimination.

To solve the system of equations using Gaussian elimination, we need to follow these steps:

### Step 1: Write the system as an augmented matrix
The given system of equations is:
\[
\begin{aligned}
2x + 3y + 4z &= 11 \quad \text{(Equation 1)}\\
x + 5y + 7z &= 15 \quad \text{(Equation 2)}\\
3x + 11y + 13z &= 25 \quad \text{(Equation 3)}
\end{aligned}
\]

The corresponding augmented matrix is:
\[
\begin{pmatrix}
2 & 3 & 4 & | & 11 \\
1 & 5 & 7 & | & 15 \\
3 & 11 & 13 & | & 25
\end{pmatrix}
\]

### Step 2: Perform row operations to get the matrix in row-echelon form
1. **Make the element in the first row, first column (pivot) equal to 1**:
   We can swap Row 1 (R1) and Row 2 (R2):
   \[
   \begin{pmatrix}
   1 & 5 & 7 & | & 15 \\
   2 & 3 & 4 & | & 11 \\
   3 & 11 & 13 & | & 25
   \end{pmatrix}
   \]

2. **Eliminate the first column elements below the pivot**:
   - Subtract 2 times Row 1 from Row 2 (R2 = R2 - 2R1):
   \[
   \begin{pmatrix}
   1 & 5 & 7 & | & 15 \\
   0 & -7 & -10 & | & -19 \\
   3 & 11 & 13 & | & 25
   \end{pmatrix}
   \]
   - Subtract 3 times Row 1 from Row 3 (R3 = R3 - 3R1):
   \[
   \begin{pmatrix}
   1 & 5 & 7 & | & 15 \\
   0 & -7 & -10 & | & -19 \\
   0 & -4 & -8 & | & -20
   \end{pmatrix}
   \]

3. **Make the pivot in the second row, second column equal to 1**:
   Divide Row 2 by -7:
   \[
   \begin{pmatrix}
   1 & 5 & 7 & | & 15 \\
   0 & 1 & \frac{10}{7} & | & \frac{19}{7} \\
   0 & -4 & -8 & | & -20
   \end{pmatrix}
   \]

4. **Eliminate the second column element below the pivot**:
   Add 4 times Row 2 to Row 3 (R3 = R3 + 4R2):
   \[
   \begin{pmatrix}
   1 & 5 & 7 & | & 15 \\
   0 & 1 & \frac{10}{7} & | & \frac{19}{7} \\
   0 & 0 & \frac{2}{7} & | & \frac{23}{7}
   \end{pmatrix}
   \]

5. **Make the pivot in the third row, third column equal to 1**:
   Multiply Row 3 by \(\frac{7}{2}\):
   \[
   \begin{pmatrix}
   1 & 5 & 7 & | & 15 \\
   0 & 1 & \frac{10}{7} & | & \frac{19}{7} \\
   0 & 0 & 1 & | & \frac{23}{2}
   \end{pmatrix}
   \]

### Step 3: Perform back-substitution to find the solution
1. **Start with the last row**:
   \[
   z = \frac{23}{2}
   \]

2. **Substitute \(z = \frac{23}{2}\) into the second row**:
   \[
   y + \frac{10}{7}\left(\frac{23}{2}\right) = \frac{19}{7}
   \]
   Simplifying:
   \[
   y + \frac{115}{14} = \frac{19}{7} = \frac{38}{14}
   \]
   \[
   y = \frac{38}{14} - \frac{115}{14} = \frac{-77}{14} = -\frac{11}{2}
   \]

3. **Substitute \(z = \frac{23}{2}\) and \(y = -\frac{11}{2}\) into the first row**:
   \[
   x + 5\left(-\frac{11}{2}\right) + 7\left(\frac{23}{2}\right) = 15
   \]
   Simplifying:
   \[
   x - \frac{55}{2} + \frac{161}{2} = 15
   \]
   \[
   x + \frac{106}{2} = 15
   \]
   \[
   x + 53 = 15
   \]
   \[
   x = 15 - 53 = -38
   \]

### Final Solution:
\[
x = -38, \quad y = -\frac{11}{2}, \quad z = \frac{23}{2}
\]

Would you like more details or have any questions? Here are 5 related questions you might consider:

1. How would the solution change if the system had infinite solutions?
2. What are the implications of a singular matrix in Gaussian elimination?
3. Can Gaussian elimination be used to solve non-square systems of equations?
4. How would you apply Gaussian elimination in solving systems with more than three variables?
5. What is the role of pivot elements in Gaussian elimination?

**Tip:** Always check the consistency of the system by substituting the solutions back into the original equations.

Learn how to solve the system of equations 2x + 3y + 4z = 11, x + 5y + 7z = 15, 3x + 11y + 13z = 25 using Gaussian elimination. Detailed step-by-step solution provided. Suitable for advanced high school students and anyone studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation