Math Problem Statement
Write an augmented matrix and use elementary row operations to solve the given system of equations: -4x + 3y = 2 and 6x - 4y = -1.
Solution
To solve the system of equations by using an augmented matrix and row reduction, here are the steps:
1. Write the augmented matrix
The system of equations is:
The augmented matrix for this system is:
-4 & 3 & | & 2 \\ 6 & -4 & | & -1 \end{bmatrix}$$ ### 2. Perform row operations to achieve reduced row-echelon form We will now perform row operations to simplify this matrix: #### Step 1: Scale Row 1 Divide Row 1 by $$-4$$ to make the leading coefficient $$1$$: $$\begin{bmatrix} 1 & -\frac{3}{4} & | & -\frac{1}{2} \\ 6 & -4 & | & -1 \end{bmatrix}$$ #### Step 2: Eliminate the first entry in Row 2 Subtract $$6 \times$$ Row 1 from Row 2: $$\begin{bmatrix} 1 & -\frac{3}{4} & | & -\frac{1}{2} \\ 0 & \frac{1}{2} & | & 2 \end{bmatrix}$$ #### Step 3: Scale Row 2 Multiply Row 2 by $$2$$ to make the second leading coefficient $$1$$: $$\begin{bmatrix} 1 & -\frac{3}{4} & | & -\frac{1}{2} \\ 0 & 1 & | & 4 \end{bmatrix}$$ #### Step 4: Eliminate the second entry in Row 1 Add $$\frac{3}{4} \times$$ Row 2 to Row 1: $$\begin{bmatrix} 1 & 0 & | & 2 \\ 0 & 1 & | & 4 \end{bmatrix}$$ ### 3. Interpret the result This matrix corresponds to the equations: $$x = 2, \; y = 4$$ ### Final Answer The solution to the system is: $$x = 2, \; y = 4$$ --- Would you like me to clarify any step? Here are five follow-up questions for further exploration: 1. How do we verify solutions for systems of linear equations? 2. What is the difference between row echelon form and reduced row echelon form? 3. Could this method be extended to larger systems of equations? If yes, how? 4. What happens if the system of equations has no solutions or infinite solutions? 5. How do row operations ensure that the solutions to the system remain unchanged? **Tip:** Always double-check your row operations to avoid arithmetic errors while solving systems of equations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Row Reduction
Matrix Operations
Formulas
Augmented Matrix Representation
Elementary Row Operations
Theorems
Row Echelon Form
Reduced Row Echelon Form
Suitable Grade Level
Grades 10-12