Math Problem Statement
The matrix
1 −1 2
0 0 0 is in reduced row-echelon form. Determine the solution of the corresponding system of linear equations. Write the solution as an ordered pair. For parametric solutions use either x = t or y = t as the parameter. (If an answer does not exist, enter DNE.) (x, y) =
Solution
The matrix given is:
1 & -1 & 2 \\ 0 & 0 & 0 \end{bmatrix}$$ This represents a system of linear equations. Each row corresponds to an equation in terms of $$x$$ and $$y$$. ### Step 1: Interpret the first row The first row is: $$1x - 1y = 2$$ or $$x - y = 2$$ ### Step 2: Solve for $$x$$ Rearranging the equation: $$x = y + 2$$ ### Step 3: Parametric form Since there is no second equation (the second row is all zeros), $$y$$ can take any value. Let $$y = t$$, where $$t$$ is a parameter. Then: $$x = t + 2$$ ### Solution as an ordered pair: $$(x, y) = (t + 2, t)$$ Thus, the solution to the system is $$(x, y) = (t + 2, t)$$, where $$t$$ is a free parameter. Would you like further details or have any questions? Here are 5 related questions to consider: 1. How do we determine if a matrix is in reduced row-echelon form? 2. What does it mean for a system of equations to have infinitely many solutions? 3. Can you generalize this method for larger systems of equations? 4. How can you graph the parametric solution $$(x, y) = (t + 2, t)$$? 5. What are the implications of having a row of zeros in a matrix? **Tip:** In systems with free variables, introducing a parameter like $$t$$ helps to describe the infinitely many solutions systematically.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Parametric Solutions
Formulas
Row-Echelon Form
Parametric Equation x = y + c
Theorems
Rank-Nullity Theorem
Free Variable Theorem
Suitable Grade Level
Grades 9-12