Answer both questions in your own words. Write at least TWO complete sentences for each question. Use the RARE method to state your answers clearly and provide evidence. What does it mean for a linear equation to have infinitely many solutions? When solving an equation, how can you tell it will have infinitely many solutions?

Question

GPT-4o mini · Answer

A linear equation having infinitely many solutions means that there are countless pairs of values that satisfy the equation. This typically occurs when the equation represents the same line in a geometric sense, meaning that each point on the line is a solution to the equation, resulting in an infinite number of solutions along that line.

You can determine that an equation will have infinitely many solutions by observing the relationships between the coefficients and constants in the equation. For example, if you manipulate the equation and end up with a statement that is always true (like 0=0) after simplifying, it indicates that any value of the variable will satisfy the equation, thus confirming that there are infinitely many solutions.