What does it mean to solve an equation, have the variables cancel out, and have the answer be 3=3, for example?
When you solve an equation, you manipulate the given equation to determine the values of the variables that satisfy the equation. The primary goal is to isolate the variable on one side of the equation.
Having the variables "cancel out" means that you can simplify the equation by performing operations that eliminate the variables from the equation. This process is usually done by applying inverse operations to both sides of the equation.
In the example you mentioned, "3=3," it indicates that both sides of the equation are equal. This equation indicates that any value for the variables would satisfy it. Therefore, there are infinitely many solutions, meaning any value for the variables that you substitute into the equation would make it true.
When we solve an equation, we aim to determine the values of the variables that make the equation true. In the process of solving, we use various algebraic operations to simplify the equation until we isolate the variable.
If all the variables cancel out during the solving process, and we end up with a statement that is always true, such as 3=3, it means that the equation is an identity. An identity is an equation that holds true for all possible values of the variables.
In the example you provided, 3=3, it means that no matter what values the variables had in the original equation, the equation will always be true. So, the solution to this equation is that it is true for all possible values.