
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.( 5 sentence paragraph)

When solving a system of equations using substitution, arriving at a statement like \( 7 = 7 \) indicates that the equations are dependent, meaning they represent the same line in a two-dimensional graph. This implies that there are infinitely many solutions, as every point on the line satisfies both equations. Graphically, this scenario reflects a situation where the two lines overlap entirely rather than intersecting at a single point. Thus, rather than a unique solution or no solution at all, the system has a solution set that includes every point along the line. This is a hallmark of systems where the equations are not only consistent but also dependent. 

Would you like more details or have any questions about this topic? Here are five related questions for further exploration:

1. What does it mean for two equations to be independent or inconsistent?
2. How can you visually identify dependent equations on a graph?
3. What are some other methods to solve systems of equations besides substitution?
4. Can you provide an example of a system that results in a unique solution?
5. How does the concept of parallel lines relate to systems of equations? 

Tip: Always check for consistency in systems by substituting values back into the original equations.

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.

Discover what it means when solving a system of equations leads to a statement like 7 = 7, indicating infinite solutions. This page covers key concepts in algebra related to dependent systems and their graphical representations, suitable for Grades 9-12.

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