If you use the substitution method and end up with the equation $2=5$, what can you say about the solutions?
The system is inconsistent.
The system is independent.
The system is dependent.
The system has an infinite number of solutions.
Answer (2)
A contradiction like 2 = 5 arises when the system of equations has no solution.
This means the equations are inconsistent, as they cannot be simultaneously true for any variable values.
Therefore, the system is inconsistent.
The final answer is: The system is inconsistent.
Explanation
Understanding the Contradiction When solving a system of equations using the substitution method, if we arrive at a contradiction like 2 = 5 , it means that the original equations are inconsistent. In simpler terms, there are no values for the variables that can satisfy both equations simultaneously.
Implication of No Solution The equation 2 = 5 is a false statement, regardless of the values of the variables. This indicates that the system of equations has no solution.
Identifying the System Type A system of equations with no solution is called an inconsistent system.
Conclusion Therefore, the correct answer is that the system is inconsistent.
Examples
Imagine you're trying to find the intersection of two lines on a graph. If the lines are parallel, they never intersect, meaning there's no solution that satisfies both equations simultaneously. Similarly, in real life, if you have two conflicting requirements (like needing to be in two places at once), you'll find no solution that meets both, representing an inconsistent system.
The equation 2 = 5 indicates a contradiction in the system of equations, meaning that there are no values for the variables that can satisfy both equations. Therefore, the system is inconsistent. The correct answer is: The system is inconsistent.
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