Math Problem Statement
Answer like previous question
Analyze two augmented matrices and select the correct solution from the multiple-choice options provided.
Solution
Let's solve these problems by analyzing the augmented matrices presented.
Question 3:
The augmented matrix is:
1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]$$ This represents the system of equations: $$x = 3$$ $$y = 4$$ $$0 = 0 \quad \text{(this equation is always true)}$$ The third row indicates that there is no constraint on $$z$$, meaning that $$z$$ is a free variable. Therefore, the system has **infinitely many solutions** where $$x = 3$$, $$y = 4$$, and $$z$$ can take any value. So the correct answer for Question 3 is: - **B. Infinitely many solutions** ### Question 4: The augmented matrix is: $$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & -2 \\ \end{array}\right]$$ This represents the system of equations: $$x = 0$$ $$y = -4$$ $$z = -2$$ Each equation defines a unique value for $$x$$, $$y$$, and $$z$$, meaning the system has a unique solution. So the correct answer for Question 4 is: - **E. Unique solution: $$x = 0$$, $$y = -4$$, $$z = -2$$** --- Do you want more details or have any further questions? Here are 5 related questions to consider: 1. How can you tell if a system of equations has no solutions from an augmented matrix? 2. What does a row of all zeros in an augmented matrix indicate? 3. How do you interpret free variables in a system of linear equations? 4. What is the difference between consistent and inconsistent systems of equations? 5. How can Gaussian elimination help solve systems of linear equations? **Tip:** If the last row of an augmented matrix becomes $$[0 \ 0 \ 0 | c]$$ where $$c \neq 0$$, the system has **no solution** (inconsistent).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Representation
Formulas
Augmented matrix
Gaussian elimination
Theorems
Existence and Uniqueness Theorem
Suitable Grade Level
Grades 10-12