When you solve an equation and end up with something like \(3 = 3\), it means you have arrived at a tautology, or an identity, which is a statement that is always true regardless of the value of the variable. This generally happens when the variables effectively cancel out during the solving process.
Here's a breakdown of what this means:
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Canceling Out Variables: In some cases, when manipulating an equation, the variables on both sides may eliminate each other. For example, consider the equation \(2x + 5 = 2x + 3 + 2\). When simplifying, you will subtract \(2x\) from both sides, yielding \(5 = 5\).
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Tautological Result: The result \(3 = 3\) indicates that the original equation had infinitely many solutions. This means any value you substitute for the variable(s) would satisfy the original equation.
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Implication: When you find an identity like \(3 = 3\), it signals that the original equation is fundamentally true for all values of the variables involved. You can think of it as saying that the relationship expressed by the original equation (before simplification) does not depend on specific values of the variables—they are always linked in such a way that their equality holds true.
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Different from No Solution: This outcome is different from getting a statement like \(5 = 7\), which would indicate that there are no solutions to the equation, meaning there is no value of the variable that could make both sides equal.
In summary, reaching a result like \(3 = 3\) when solving an equation indicates the equation is an identity and has infinitely many solutions. It suggests that the variable involved can take on any value while still satisfying the equation.
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