Math Problem Statement
Below are augmented matrices (in reduced row echelon form) corresponding to the systems of linear equations. Determine if each system is consistent with a unique solution, consistent with infinitely many solutions or inconsistent: [(1,0,-9,2),(0,1,-1,-6),(0,0,0,0)]
Solution
Let's analyze the augmented matrix:
1 & 0 & -9 & 2 \\ 0 & 1 & -1 & -6 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$ This is the reduced row echelon form (RREF) of the augmented matrix. The matrix represents the following system of equations: 1. $$x_1 - 9x_3 = 2$$ 2. $$x_2 - x_3 = -6$$ 3. $$0 = 0$$ (which is always true and doesn't add any new information) ### Step 1: Analyze the system - There are three unknowns: $$x_1$$, $$x_2$$, and $$x_3$$. - There are two non-trivial equations (from the first two rows), while the third row gives $$0 = 0$$, which is trivially true and doesn't affect the solution space. ### Step 2: Determine if the system is consistent - The system is consistent because there is no row that corresponds to an equation like $$0 = c$$, where $$c$$ is a non-zero constant (which would indicate inconsistency). - The presence of a free variable (since $$x_3$$ is not a leading variable) suggests that there are infinitely many solutions. ### Step 3: Identify the type of solution - $$x_3$$ is a free variable because it does not have a leading 1 in the matrix, meaning we can assign any value to $$x_3$$. - We can express $$x_1$$ and $$x_2$$ in terms of $$x_3$$. Thus, the system has **infinitely many solutions**, as there is at least one free variable (in this case, $$x_3$$). ### Conclusion: The system is consistent with infinitely many solutions. --- Would you like details on how to parametrize the solutions or any further explanation? Here are 5 related questions: 1. How do you identify free variables in a system of equations? 2. What makes a system of equations inconsistent? 3. How do you find the general solution of a system with infinitely many solutions? 4. What is the difference between row echelon form and reduced row echelon form? 5. Can a system have both a unique and infinite solution? **Tip:** In RREF, any row that is completely zero indicates that the system has either infinitely many solutions or no solution, depending on other rows.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Row Echelon Form
Infinitely Many Solutions
Formulas
-
Theorems
Consistency of Systems of Linear Equations
Free Variables Theorem
Suitable Grade Level
Grades 10-12 or College Level
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