Math Problem Statement
solve the following system of equations using matrices and elementary row operations. x+y=6, 2x+4y+z=5, 2x+3y+z=6
Solution
To solve the system of equations using matrices and elementary row operations, follow these steps:
System of Equations:
Step 1: Write the augmented matrix
The augmented matrix for this system is:
1 & 1 & 0 & 6 \\ 2 & 4 & 1 & 5 \\ 2 & 3 & 1 & 6 \\ \end{bmatrix}$$ ### Step 2: Perform Elementary Row Operations 1. **Make the first pivot 1**: The pivot in the first row is already 1. 2. **Eliminate the first column below the pivot**: - Subtract $$2 \times \text{Row 1}$$ from Row 2: $$R2 = R2 - 2R1$$ - Subtract $$2 \times \text{Row 1}$$ from Row 3: $$R3 = R3 - 2R1$$ Resulting matrix: $$\begin{bmatrix} 1 & 1 & 0 & 6 \\ 0 & 2 & 1 & -7 \\ 0 & 1 & 1 & -6 \\ \end{bmatrix}$$ 3. **Make the second pivot 1**: Divide Row 2 by 2: $$R2 = \frac{R2}{2}$$ Resulting matrix: $$\begin{bmatrix} 1 & 1 & 0 & 6 \\ 0 & 1 & \frac{1}{2} & -\frac{7}{2} \\ 0 & 1 & 1 & -6 \\ \end{bmatrix}$$ 4. **Eliminate the second column below and above the pivot**: - Subtract Row 2 from Row 3: $$R3 = R3 - R2$$ - Subtract $$1 \times \text{Row 2}$$ from Row 1: $$R1 = R1 - R2$$ Resulting matrix: $$\begin{bmatrix} 1 & 0 & -\frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{1}{2} & -\frac{7}{2} \\ 0 & 0 & \frac{1}{2} & \frac{1}{2} \\ \end{bmatrix}$$ 5. **Make the third pivot 1**: Multiply Row 3 by 2: $$R3 = 2R3$$ Resulting matrix: $$\begin{bmatrix} 1 & 0 & -\frac{1}{2} & \frac{5}{2} \\ 0 & 1 & \frac{1}{2} & -\frac{7}{2} \\ 0 & 0 & 1 & 1 \\ \end{bmatrix}$$ 6. **Eliminate the third column above the pivot**: - Add $$\frac{1}{2} \times \text{Row 3}$$ to Row 1: $$R1 = R1 + \frac{1}{2}R3$$ - Subtract $$\frac{1}{2} \times \text{Row 3}$$ from Row 2: $$R2 = R2 - \frac{1}{2}R3$$ Resulting matrix: $$\begin{bmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 1 \\ \end{bmatrix}$$ ### Step 3: Interpret the Solution The final augmented matrix corresponds to the equations: $$x = 3, \; y = -4, \; z = 1$$ ### Final Solution: $$\boxed{x = 3, \; y = -4, \; z = 1}$$ Would you like more details on the process or further explanations? --- **Related Questions:** 1. How do you verify solutions for a system of equations? 2. Can systems of equations have no solutions or infinite solutions? 3. How does row reduction differ from other methods like substitution? 4. What are real-world applications of solving systems of equations? 5. How does matrix rank relate to solving systems of equations? **Tip:** When solving systems, always double-check your augmented matrix setup to avoid mistakes in the row operations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Row Reduction
Formulas
Augmented Matrix Representation
Elementary Row Operations (Row Addition, Scaling, Row Replacement)
Theorems
Gaussian Elimination
Row Echelon Form
Suitable Grade Level
Grades 9-12
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