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.

Answer 1

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.