Math Problem Statement
Solve for 14.
Solve for problem 14.
Solution
To solve problem 14 using augmented matrices, we represent the system of equations as a matrix and apply Gaussian elimination.
Problem 14:
The equations are:
Step 1: Write the augmented matrix
2 & 6 & 8 & | & 5 \\ -2 & 9 & -12 & | & -1 \\ 4 & 6 & -4 & | & 3 \end{bmatrix}$$ ### Step 2: Perform Gaussian elimination #### First, normalize the first row: Divide the first row by 2: $$\begin{bmatrix} 1 & 3 & 4 & | & 2.5 \\ -2 & 9 & -12 & | & -1 \\ 4 & 6 & -4 & | & 3 \end{bmatrix}$$ #### Eliminate the first entry in the second and third rows: For row 2: $$R_2 = R_2 + 2R_1$$ For row 3: $$R_3 = R_3 - 4R_1$$ Result: $$\begin{bmatrix} 1 & 3 & 4 & | & 2.5 \\ 0 & 15 & -4 & | & 4 \\ 0 & -6 & -20 & | & -7 \end{bmatrix}$$ #### Normalize row 2: Divide row 2 by 15: $$\begin{bmatrix} 1 & 3 & 4 & | & 2.5 \\ 0 & 1 & -\frac{4}{15} & | & \frac{4}{15} \\ 0 & -6 & -20 & | & -7 \end{bmatrix}$$ #### Eliminate the second entry in row 3: For row 3: $$R_3 = R_3 + 6R_2$$ Result: $$\begin{bmatrix} 1 & 3 & 4 & | & 2.5 \\ 0 & 1 & -\frac{4}{15} & | & \frac{4}{15} \\ 0 & 0 & -\frac{76}{5} & | & -\frac{17}{5} \end{bmatrix}$$ #### Normalize row 3: Divide row 3 by $$-\frac{76}{5}$$: $$\begin{bmatrix} 1 & 3 & 4 & | & 2.5 \\ 0 & 1 & -\frac{4}{15} & | & \frac{4}{15} \\ 0 & 0 & 1 & | & \frac{17}{76} \end{bmatrix}$$ #### Back substitution: 1. Use row 3 to eliminate $$z$$ in rows 1 and 2. 2. Use row 2 to eliminate $$y$$ in row 1. After back substitution, we find: $$x = \frac{3}{2}, \, y = \frac{1}{2}, \, z = \frac{17}{76}.$$ ### Solution: $$x = \frac{3}{2}, \, y = \frac{1}{2}, \, z = \frac{17}{76}.$$ Let me know if you'd like detailed steps for back substitution or have any questions! #### Related Questions: 1. How does Gaussian elimination differ from back substitution? 2. What are other methods to solve systems of linear equations? 3. Can you verify the solution by substituting $$x, y, z$$ back into the equations? 4. What is the importance of augmented matrices in solving equations? 5. How do row operations impact the determinant of a matrix? **Tip:** Always ensure your leading coefficient (pivot) is 1 before proceeding with elimination.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Gaussian Elimination
Formulas
Gaussian elimination steps for augmented matrices
Theorems
Row reduction theorem for solving linear systems
Suitable Grade Level
Grades 10-12