Math Problem Statement
Solve the system of equations using Gaussian elimination:
1. y + 3z = 4
2. -2x + 3y + 13z = 10
3. -x + y + 5z = 3.
Solution
To solve the given system of equations using Gaussian elimination, we will first rewrite the system into an augmented matrix and then row reduce it.
The system is:
1. \ & y + 3z = 4, \\ 2. \ & -2x + 3y + 13z = 10, \\ 3. \ & -x + y + 5z = 3. \end{aligned}$$ ### Step 1: Write the augmented matrix $$\begin{bmatrix} 0 & 1 & 3 & 4 \\ -2 & 3 & 13 & 10 \\ -1 & 1 & 5 & 3 \end{bmatrix}$$ ### Step 2: Perform Gaussian elimination #### First, eliminate the first column. 1. Swap Row 1 and Row 2 (so that the pivot in the first row is non-zero): $$\begin{bmatrix} -2 & 3 & 13 & 10 \\ 0 & 1 & 3 & 4 \\ -1 & 1 & 5 & 3 \end{bmatrix}$$ 2. Divide Row 1 by -2 to make the pivot 1: $$\begin{bmatrix} 1 & -\frac{3}{2} & -\frac{13}{2} & -5 \\ 0 & 1 & 3 & 4 \\ -1 & 1 & 5 & 3 \end{bmatrix}$$ 3. Add Row 1 to Row 3 (to eliminate the first element in Row 3): $$\begin{bmatrix} 1 & -\frac{3}{2} & -\frac{13}{2} & -5 \\ 0 & 1 & 3 & 4 \\ 0 & \frac{-1}{2} & \frac{-3}{2} & -2 \end{bmatrix}$$ #### Next, eliminate the second column. 1. Add $$\frac{1}{2}$$ of Row 2 to Row 3 (to eliminate the second element in Row 3): $$\begin{bmatrix} 1 & -\frac{3}{2} & -\frac{13}{2} & -5 \\ 0 & 1 & 3 & 4 \\ 0 & 0 & -1 & 0 \end{bmatrix}$$ 2. Divide Row 3 by -1 to make the pivot 1: $$\begin{bmatrix} 1 & -\frac{3}{2} & -\frac{13}{2} & -5 \\ 0 & 1 & 3 & 4 \\ 0 & 0 & 1 & 0 \end{bmatrix}$$ #### Back substitution: From Row 3: $$z = 0$$. From Row 2: $$y + 3z = 4 \Rightarrow y = 4$$. From Row 1: $$x - \frac{3}{2}y - \frac{13}{2}z = -5 \Rightarrow x - \frac{3}{2}(4) = -5 \Rightarrow x = 1$$. ### Final Answer: The system is consistent and independent: $$x = 1, \ y = 4, \ z = 0.$$ Would you like a step-by-step walkthrough or further clarification on Gaussian elimination? --- Here are some related questions to enhance your understanding: 1. How can you determine if a system of equations is consistent or inconsistent? 2. What are the differences between Gaussian elimination and Gauss-Jordan elimination? 3. How does swapping rows affect the determinant of the coefficient matrix? 4. Can you identify cases where a system has infinitely many solutions? 5. What is the geometric interpretation of a consistent and independent system? **Tip:** When solving systems of equations, always double-check row operations to avoid calculation errors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Systems of Equations
Formulas
Gaussian elimination method
Theorems
Gaussian elimination
Suitable Grade Level
Grades 10-12