When you solve an equation and end up with something like \(3 = 3\) after simplifying (and possibly canceling out variables), it means that the original equation is an identity. An identity is an equation that is true for all values of the variables involved.
Here's a breakdown of what this means:
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Equivalence: If you have transformed the equation all the way down to \(3 = 3\), it indicates that the two sides of the original equation are equivalent for any value of the variable(s). No matter what value you substitute for the variable(s), the equation will always hold true.
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Infinite Solutions: Since the equation is always true, it implies that there are infinitely many solutions. In other words, any number you can plug in for the variable(s) will satisfy the original equation.
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Example Scenario: Consider the equation \(2x + 4 = 2(x + 2)\). If you solve it, you might simplify it step by step:
- Distributing on the right side gives \(2x + 4 = 2x + 4\).
- Now if you subtract \(2x\) from both sides, you get \(4 = 4\).
- This result (\(4 = 4\)) tells you that the equation is true for all values of \(x\), meaning any \(x\) is a solution.
In summary, solving an equation and arriving at a statement like \(3 = 3\) indicates that the equation is valid for all possible values of the variables involved, leading to an infinite set of solutions.