Math Problem Statement
Solution
We are given two systems of linear equations, and we need to determine which system has no solution and which one has infinitely many solutions without graphing them.
System 1:
5x + y &= 13 \\ 20x + 4y &= 64 \end{aligned}$$ ### System 2: $$\begin{aligned} 5x + y &= 13 \\ 20x &= 52 - 4y \end{aligned}$$ ### Analyzing System 1: 1. The first equation is $$5x + y = 13$$. 2. The second equation is $$20x + 4y = 64$$. Notice that this is simply the first equation multiplied by 4: $$4(5x + y) = 4(13) \implies 20x + 4y = 52$$ But the second equation is $$20x + 4y = 64$$, not 52. Therefore, the two equations are inconsistent. This means **System 1 has no solution**. ### Analyzing System 2: 1. The first equation is $$5x + y = 13$$. 2. Rewriting the second equation: $$20x = 52 - 4y \implies 20x + 4y = 52$$ This is essentially the same as the first equation multiplied by 4. So, the second equation is just a rearranged form of the first. Since both equations are equivalent, there are **infinitely many solutions** for System 2. ### Conclusion: - **System 1** has no solution (inconsistent equations). - **System 2** has infinitely many solutions (dependent equations). Would you like further details on how to analyze systems of linear equations? Here are 5 related questions to expand your understanding: 1. How can you tell when two systems of equations are inconsistent without solving them? 2. What are the methods to solve a system of linear equations algebraically? 3. How does the elimination method help in identifying no solution or infinitely many solutions? 4. Can two linear equations with the same slope have a unique solution? 5. How does the concept of parallel lines relate to systems of equations? **Tip:** When analyzing systems of linear equations, always check if one equation is a multiple of the other. This can help you identify if there are infinitely many solutions or no solution at all.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems
Consistent and Inconsistent Systems
Dependent and Independent Equations
Formulas
5x + y = 13
20x + 4y = 64
20x = 52 - 4y
Theorems
The concept of inconsistent systems (parallel lines, no intersection)
The concept of dependent systems (same line, infinitely many solutions)
Suitable Grade Level
Grades 8-10
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