Math Problem Statement
Solve the system of equations: x√2 - 2y = √2 and x - y√2 = 2
Solution
The problem you provided involves solving the following system of linear equations:
x\sqrt{2} - 2y = \sqrt{2} \\ x - y\sqrt{2} = 2 \end{cases}$$ Let's solve this system step by step: ### Step 1: Express $$x$$ from the second equation From the second equation $$x - y\sqrt{2} = 2$$, we can isolate $$x$$: $$x = y\sqrt{2} + 2$$ ### Step 2: Substitute $$x$$ into the first equation Now substitute $$x = y\sqrt{2} + 2$$ into the first equation: $$(y\sqrt{2} + 2)\sqrt{2} - 2y = \sqrt{2}$$ Distribute $$\sqrt{2}$$ on the left side: $$y(\sqrt{2})^2 + 2\sqrt{2} - 2y = \sqrt{2}$$ Since $$(\sqrt{2})^2 = 2$$, this simplifies to: $$2y + 2\sqrt{2} - 2y = \sqrt{2}$$ ### Step 3: Simplify and solve for $$y$$ The $$2y$$ terms cancel out, leaving: $$2\sqrt{2} = \sqrt{2}$$ This is a contradiction because $$2\sqrt{2} \neq \sqrt{2}$$. ### Conclusion: The system has no solution because it leads to a contradiction. Would you like more details or have any questions about this process? Here are five related questions you can explore: 1. How do we solve a system of equations using substitution in general? 2. What does it mean if solving a system leads to a contradiction? 3. How can we use matrices to solve systems of linear equations? 4. What are some common types of systems of equations (e.g., consistent, inconsistent)? 5. How does the value of square roots affect the solution of algebraic equations? **Tip**: Always check if a system of equations leads to a contradiction or has infinite solutions before concluding the result.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Systems of Equations
Algebra
Square Roots
Formulas
Substitution method
Square root properties
Theorems
Inconsistency in systems of equations
Suitable Grade Level
Grades 9-12