Math Problem Statement
Solution
The image shows two systems of equations (System A and System B), and the goal is to determine the type of solution for each system: whether the system has no solution, a unique solution, or infinitely many solutions.
Let’s analyze each system.
System A:
x + 5y = 5$$ 1. Add the two equations together to eliminate $$x$$: $$(-x - 5y) + (x + 5y) = 5 + 5 \implies 0 = 10$$ This is a contradiction, meaning System A has **no solution**. ### System B: $$x + 4y = 8 \\ -x - 4y = -8$$ 1. Add the two equations: $$(x + 4y) + (-x - 4y) = 8 + (-8) \implies 0 = 0$$ This is a true statement, meaning that the two equations are actually the same, just multiplied by -1. Therefore, System B has **infinitely many solutions**, and the solutions must satisfy one of the equations. Solving for $$y$$ in terms of $$x$$, from the first equation: $$x + 4y = 8 \implies 4y = 8 - x \implies y = \frac{8 - x}{4}$$ Thus, the results are: - **System A**: No solution. - **System B**: Infinitely many solutions, with $$y = \frac{8 - x}{4}$$. Would you like further details or clarifications? Here are 5 related questions you could explore: 1. How can you determine when a system of equations has no solution? 2. What is the difference between dependent and independent systems of equations? 3. How do you solve systems of equations using substitution or elimination methods? 4. What does it mean geometrically when two lines represent a system with infinitely many solutions? 5. How can you recognize a system of equations with infinitely many solutions by inspection? **Tip:** When two equations in a system result in a contradiction like $$0 = 10$$, it indicates that the lines are parallel, meaning no solution exists.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Elimination Method
Formulas
Add equations to eliminate variables
Solve for y: y = (8 - x)/4
Theorems
A system has no solution if the result is a contradiction
A system has infinitely many solutions if the result is a true statement like 0 = 0
Suitable Grade Level
Grades 8-10
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