Math Problem Statement
4x+3y+z=14 x-3y+2z=-6 11x-2y+3z=8 the solution set is {}
Solution
We can solve this system of three linear equations using either substitution, elimination, or matrix methods. In this case, I will use Gaussian elimination to find the solution set.
The system of equations is:
1. & \quad 4x + 3y + z = 14 \\ 2. & \quad x - 3y + 2z = -6 \\ 3. & \quad 11x - 2y + 3z = 8 \end{aligned}$$ ### Step 1: Write the system as an augmented matrix. $$\begin{pmatrix} 4 & 3 & 1 & | & 14 \\ 1 & -3 & 2 & | & -6 \\ 11 & -2 & 3 & | & 8 \\ \end{pmatrix}$$ ### Step 2: Perform row operations to achieve row echelon form. 1. Swap **Row 1** and **Row 2** to get a leading 1 in the first position. $$\begin{pmatrix} 1 & -3 & 2 & | & -6 \\ 4 & 3 & 1 & | & 14 \\ 11 & -2 & 3 & | & 8 \\ \end{pmatrix}$$ 2. Eliminate the first element of rows 2 and 3 by subtracting suitable multiples of Row 1. - Row 2 → Row 2 - 4(Row 1) - Row 3 → Row 3 - 11(Row 1) $$\begin{pmatrix} 1 & -3 & 2 & | & -6 \\ 0 & 15 & -7 & | & 38 \\ 0 & 31 & -19 & | & 74 \\ \end{pmatrix}$$ ### Step 3: Make the pivot in the second column a 1 by dividing Row 2 by 15. $$\begin{pmatrix} 1 & -3 & 2 & | & -6 \\ 0 & 1 & -\frac{7}{15} & | & \frac{38}{15} \\ 0 & 31 & -19 & | & 74 \\ \end{pmatrix}$$ ### Step 4: Eliminate the second column of Row 3 by subtracting 31 times Row 2 from Row 3. $$\begin{pmatrix} 1 & -3 & 2 & | & -6 \\ 0 & 1 & -\frac{7}{15} & | & \frac{38}{15} \\ 0 & 0 & \frac{14}{15} & | & \frac{32}{15} \\ \end{pmatrix}$$ ### Step 5: Solve for $$z$$, $$y$$, and $$x$$. - From the third row: $$\frac{14}{15}z = \frac{32}{15} \quad \Rightarrow \quad z = \frac{32}{14} = \frac{16}{7}$$ - Substitute $$z = \frac{16}{7}$$ into Row 2 to solve for $$y$$: $$y - \frac{7}{15} \times \frac{16}{7} = \frac{38}{15} \quad \Rightarrow \quad y = \frac{8}{3}$$ - Substitute $$y = \frac{8}{3}$$ and $$z = \frac{16}{7}$$ into Row 1 to solve for $$x$$: $$x - 3 \times \frac{8}{3} + 2 \times \frac{16}{7} = -6 \quad \Rightarrow \quad x = -\frac{17}{7}$$ ### Solution set: $$x = -\frac{17}{7}, \quad y = \frac{8}{3}, \quad z = \frac{16}{7}$$ Would you like further details on any of the steps? ### 5 Related Questions: 1. How is Gaussian elimination different from Gauss-Jordan elimination? 2. What are the benefits of using matrix methods for solving systems of equations? 3. How can row operations affect the solution of a system of equations? 4. Can a system of linear equations have no solution or infinite solutions? 5. How does back-substitution work in solving systems of equations? ### Tip: Always check the consistency of your system of equations before attempting to solve them, as some systems may not have a unique solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Matrix Operations
Gaussian Elimination
Formulas
Row Operations for Matrix Manipulation
Back-Substitution Method
Theorems
Gaussian Elimination
Suitable Grade Level
Grades 10-12
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