Math Problem Statement
Solve the following system of equations using Gaussian Elimination method:
1. x + y - 2z = 1
2. 2x - y + 4z = 9
3. x - 3y = 6
Solution
Let's solve the given system of equations using Gaussian Elimination Method step-by-step:
System of equations:
Step 1: Represent the system in augmented matrix form
1 & 1 & -2 & | & 1 \\ 2 & -1 & 4 & | & 9 \\ 1 & -3 & 0 & | & 6 \end{bmatrix}$$ --- ### Step 2: Perform row operations to reduce the matrix to row echelon form #### Eliminate the first entry of row 2 ($$a_{21}$$) and row 3 ($$a_{31}$$): - Subtract $$2R_1$$ from $$R_2$$: $$R_2 \to R_2 - 2R_1$$ - Subtract $$R_1$$ from $$R_3$$: $$R_3 \to R_3 - R_1$$ **New matrix:** $$\begin{bmatrix} 1 & 1 & -2 & | & 1 \\ 0 & -3 & 8 & | & 7 \\ 0 & -4 & 2 & | & 5 \end{bmatrix}$$ --- #### Eliminate the second entry of row 3 ($$a_{32}$$): - Divide $$R_2$$ by $$-3$$: $$R_2 \to R_2 / -3$$ $$\begin{bmatrix} 1 & 1 & -2 & | & 1 \\ 0 & 1 & -\frac{8}{3} & | & -\frac{7}{3} \\ 0 & -4 & 2 & | & 5 \end{bmatrix}$$ - Add $$4R_2$$ to $$R_3$$: $$R_3 \to R_3 + 4R_2$$ **New matrix:** $$\begin{bmatrix} 1 & 1 & -2 & | & 1 \\ 0 & 1 & -\frac{8}{3} & | & -\frac{7}{3} \\ 0 & 0 & -\frac{10}{3} & | & \frac{1}{3} \end{bmatrix}$$ --- ### Step 3: Back substitution to find solutions From the last row: $$-\frac{10}{3}z = \frac{1}{3} \implies z = -\frac{1}{10} \cdot 3 = -\frac{1}{10}$$ Substitute $$z = -\frac{1}{10}$$ into the second row: $$y - \frac{8}{3}z = -\frac{7}{3} \implies y - \frac{8}{3} \cdot -\frac{1}{10} = -\frac{7}{3}$$ $$y + \frac{4}{15} = -\frac{7}{3} \implies y = -\frac{7}{3} - \frac{4}{15}$$ Simplify: $$y = -\frac{35}{15} - \frac{4}{15} = -\frac{39}{15} = -\frac{13}{5}$$ Substitute $$y = -\frac{13}{5}$$ and $$z = -\frac{1}{10}$$ into the first row: $$x + y - 2z = 1 \implies x - \frac{13}{5} - 2(-\frac{1}{10}) = 1$$ $$x - \frac{13}{5} + \frac{1}{5} = 1 \implies x - \frac{12}{5} = 1$$ $$x = 1 + \frac{12}{5} = \frac{5}{5} + \frac{12}{5} = \frac{17}{5}$$ --- ### Final Solution: $$x = \frac{17}{5}, \quad y = -\frac{13}{5}, \quad z = -\frac{1}{10}$$ Would you like a detailed explanation of any step or additional examples? --- ### Related Questions: 1. What are the main advantages of using Gaussian Elimination over substitution methods? 2. How can Gaussian Elimination be applied to larger systems of equations? 3. What are some potential pitfalls or errors to watch for during Gaussian Elimination? 4. How is row reduction connected to matrix invertibility? 5. Can Gaussian Elimination always provide a solution for any system of equations? Why or why not? **Tip:** Always double-check row operations to ensure accuracy during elimination steps!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Gaussian Elimination
Linear Algebra
System of Linear Equations
Formulas
Augmented matrix representation
Row operations for matrix reduction
Back substitution method for solving equations
Theorems
Gaussian Elimination method
Suitable Grade Level
Grades 10-12 (High School) or Undergraduate Level