When Joanne substituted the value of the variable back into the equation and got 12.34 = 12.33, it indicates that she is very close to the correct answer, but there is a slight discrepancy. The reason she could still have done her work correctly, despite this slight difference, can be attributed to several factors:
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Rounding Errors: It's possible that during her calculations or when she rounded her final answer, she obtained 12.34 instead of the exact value. Many calculations involving decimals can introduce rounding errors, which can lead to minor differences in the final result.
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Approximation: Joanne might have calculated an approximate value rather than the exact solution. If the problem involved estimating or rounding numbers, it’s common for the result to be very close but not exactly correct.
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Measurement Errors: If her original values or constants were measurements that had a level of uncertainty, the result would reflect that uncertainty. For example, if she was using measured lengths that were not precise, small differences might arise when these values are substituted back into an equation.
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Algebraic Manipulation: It’s possible she arrived at an intermediate step that was correct, but when moving to the next steps, a minor error occurred, affecting the final comparison but not the overall methodology she employed.
In summary, though she didn’t get a perfectly true statement from the substitution, the fact that she came so close (12.34 being nearly equal to 12.33) suggests that her calculations were likely on the right track but suffered from slight inaccuracies or approximations. This illustrates that mathematical processes can sometimes yield results that differ by small amounts while still being otherwise valid or close to correct.
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