Question
Determine if the following system of equations has no solutions, infinitely many solutions or
exactly one solution.
Infinitely Many Solutions
No Solutions
exactly one solution.
Infinitely Many Solutions
No Solutions
Ask by Stanley Henry. in the United States
Jan 22,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The system has exactly one solution:
Solution
To determine whether the given system of equations has no solutions, infinitely many solutions, or exactly one solution, let’s analyze the system:
Step 1: Calculate the Determinant
First, consider the coefficient matrix:
The determinant (
Since the determinant
Step 2: Solve the System
Alternatively, we can solve the system using substitution or elimination.
- Multiply the first equation by 3:
- Add this to the second equation:
- Substitute
into the first equation:
Conclusion:
The system has exactly one solution:
Answer:
The system has exactly one solution.
The system has exactly one solution.
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Simplify this solution Extra Insights
To determine the nature of the solution for the given system of equations, we can manipulate one equation to express its relationship with the other. If we multiply the first equation,
Revisiting systems of equations, you can often find that parallel lines have the same slope but different y-intercepts, leading to that no single point satisfies both equations. In contrast, coincident lines (exactly the same line) yield infinitely many solutions, while intersecting lines result in a unique solution. Always check the slopes and intercepts!
