Math Problem Statement

Solve

Solution

To determine the solution of the given augmented matrix in row echelon form, we need to interpret the matrix and translate it into a system of linear equations. The given matrix is:

1 & -\frac{1}{3} & 2 \\ 0 & 0 & 0 \end{array}\right]$$ This matrix represents the following system of equations: 1. $$x - \frac{1}{3}y = 2$$ 2. $$0 = 0$$ (which is always true and does not provide any new information) From the first equation, we can solve for $$y$$ in terms of $$x$$: $$x - \frac{1}{3}y = 2 \implies y = 3x - 6$$ Thus, the solution to the system can be expressed in terms of $$x$$ (a free variable), which gives us the general solution: $$(x, y) = (x, 3x - 6)$$ Comparing this with the provided options, we find that the correct option is: 5) $$(x, 3x - 6)$$ Would you like further details or have any questions? Here are five more questions you might be interested in: 1. How do you convert a matrix into row echelon form? 2. Can you explain the concept of free variables in systems of linear equations? 3. What is the difference between row echelon form and reduced row echelon form? 4. How do you determine if a system of linear equations has no solution, one solution, or infinitely many solutions? 5. What are some real-world applications of solving systems of linear equations? **Tip:** When solving a system of linear equations, it is important to check if the system is consistent (has at least one solution) or inconsistent (has no solution). Consistency can often be determined by looking at the row echelon form of the matrix and ensuring there are no contradictory equations.

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Row Echelon Form
Free Variables

Formulas

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Theorems

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Suitable Grade Level

High School

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