Math Problem Statement
Solution
To construct the augmented matrix for this system of equations, we need to first simplify each equation and express them in standard form, . Here are the given equations:
Let's rewrite each equation step by step:
Step 1: Simplify the first equation
Rewrite as:
Step 2: Simplify the second equation
Start with: Multiply both sides by 9: Expand: Rewrite as:
Step 3: Simplify the third equation
Start with: Expand the right-hand side: Move terms to one side: This simplifies to:
Step 4: Form the augmented matrix
The system of equations in matrix form is:
The augmented matrix is:
2 & -1 & 0 & | -9 \\ -4 & 32 & 5 & | 0 \\ 2 & -1 & 12 & | 0 \\ \end{bmatrix}$$ Would you like a detailed explanation of any step or have any questions? Here are 5 related questions to consider: 1. How do you convert a system of linear equations into an augmented matrix? 2. What are the steps to solve a system of equations using matrix row reduction? 3. How does changing the form of an equation affect its representation in a matrix? 4. What does each row in an augmented matrix represent geometrically? 5. How can you tell if a system of linear equations is inconsistent using its augmented matrix? **Tip:** When forming augmented matrices, always ensure equations are in standard form $$ax + by + cz = d$$ before constructing the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
System of Linear Equations
Matrices
Formulas
Ax = b form
Matrix representation of linear systems
Theorems
Matrix Augmentation Theorem
Row Reduction
Suitable Grade Level
Undergraduate (College Level)
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