Math Problem Statement
Express the following system of equations in matrix form and solve them by the elimination method: 2x_1+x_2+2x_3+x_4=6; 6x_1-6x_2+6x_3+12x_4=36; 4x_1+4x_2+3x_3-3x_4= -1; 2x_1+2x_2-x_3+x_4=10
write anser like its from a reference book and dont include much senetences
Solution
The given system of equations can be expressed in matrix form as:
2 & 1 & 2 & 1 \\ 6 & -6 & 6 & 12 \\ 4 & 4 & 3 & -3 \\ 2 & 2 & -1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 6 \\ 36 \\ -1 \\ 10 \end{bmatrix}$$ Using the elimination method, perform row operations: 1. Divide row 1 by 2 to simplify: $$R_1 \rightarrow \frac{1}{2} R_1$$ $$\begin{bmatrix} 1 & 0.5 & 1 & 0.5 \\ 6 & -6 & 6 & 12 \\ 4 & 4 & 3 & -3 \\ 2 & 2 & -1 & 1 \end{bmatrix}$$ 2. Eliminate the first column below the pivot: $$R_2 \rightarrow R_2 - 6R_1, \quad R_3 \rightarrow R_3 - 4R_1, \quad R_4 \rightarrow R_4 - 2R_1$$ $$\begin{bmatrix} 1 & 0.5 & 1 & 0.5 \\ 0 & -9 & 0 & 9 \\ 0 & 2 & -1 & -5 \\ 0 & 1 & -3 & 0 \end{bmatrix}$$ 3. Normalize the second row by dividing by -9: $$R_2 \rightarrow \frac{1}{-9} R_2$$ $$\begin{bmatrix} 1 & 0.5 & 1 & 0.5 \\ 0 & 1 & 0 & -1 \\ 0 & 2 & -1 & -5 \\ 0 & 1 & -3 & 0 \end{bmatrix}$$ 4. Eliminate second column entries below the pivot: $$R_3 \rightarrow R_3 - 2R_2, \quad R_4 \rightarrow R_4 - R_2$$ $$\begin{bmatrix} 1 & 0.5 & 1 & 0.5 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & -1 & -3 \\ 0 & 0 & -3 & 1 \end{bmatrix}$$ 5. Normalize row 3: $$R_3 \rightarrow -R_3$$ $$\begin{bmatrix} 1 & 0.5 & 1 & 0.5 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & -3 & 1 \end{bmatrix}$$ 6. Eliminate the third column entry in row 4: $$R_4 \rightarrow R_4 + 3R_3$$ $$\begin{bmatrix} 1 & 0.5 & 1 & 0.5 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 10 \end{bmatrix}$$ 7. Solve for $$x_4$$: $$x_4 = \frac{10}{10} = 1$$ 8. Back-substituting: $$x_3 = 3(1) = 3$$ $$x_2 = -1(1) = -1$$ $$x_1 = 6 - (0.5)(-1) - (1)(3) - (0.5)(1) = 3$$ ### Solution: $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 3 \\ -1 \\ 3 \\ 1 \end{bmatrix}$$ --- Do you want additional details? ### Related Questions: 1. How does the elimination method differ from the Gauss-Jordan method? 2. Can the elimination method be used for non-square systems? 3. How do row operations affect the determinant of a matrix? 4. What happens if a system has no solution in elimination? 5. How can matrix inversion be used to solve a system of equations? **Tip:** Always check the determinant before using elimination. If it's zero, the system may be dependent or inconsistent.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Elimination Method
Matrix Operations
Formulas
Matrix form of a system of linear equations
Elimination method steps
Theorems
Gaussian elimination
Suitable Grade Level
Grades 11-12