Question
You use substitution to solve a system of equations and, after simplifying, end with a statement that says 7 = 7. Describe what this statement means about the number of solutions and about the graph of the system.
Answers
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GPT-4o mini
When you end up with a statement like \( 7 = 7 \) after solving a system of equations using substitution, it indicates that the two equations are not just equivalent, but they represent the same line in a two-dimensional space.
Here's what this means about the number of solutions and the graph of the system:
1. **Number of Solutions**: The system has infinitely many solutions. This is because every point on the line represented by one equation is also a point on the line represented by the other equation. Since they describe the same line, there is not a single solution, but rather an infinite number of points (solutions) that satisfy both equations.
2. **Graph of the System**: Graphically, when you plot the two equations, you will see that they lie on top of each other, meaning the lines coincide. This means that for every \( x \)-value on the line, there is a corresponding \( y \)-value that makes both equations true.
In summary, the \( 7 = 7 \) conclusion suggests that the system of equations is dependent and consistent, meaning it has infinitely many solutions that are represented by the same line in the graph.
Here's what this means about the number of solutions and the graph of the system:
1. **Number of Solutions**: The system has infinitely many solutions. This is because every point on the line represented by one equation is also a point on the line represented by the other equation. Since they describe the same line, there is not a single solution, but rather an infinite number of points (solutions) that satisfy both equations.
2. **Graph of the System**: Graphically, when you plot the two equations, you will see that they lie on top of each other, meaning the lines coincide. This means that for every \( x \)-value on the line, there is a corresponding \( y \)-value that makes both equations true.
In summary, the \( 7 = 7 \) conclusion suggests that the system of equations is dependent and consistent, meaning it has infinitely many solutions that are represented by the same line in the graph.