Let's analyze Carlos's solution step-by-step:
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Starting Equation: \[ 5x + 7.5 = 25 \]
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Carlos's Work: Carlos correctly started with the equation but made an error while performing inverse operations. Let's follow his solution and highlight the mistake:
- Subtract 7.5 from both sides: \[ 5x = 25 - 7.5 \]
Calculating the right-hand side: \[ 25 - 7.5 = 17.5 \] So the equation should be: \[ 5x = 17.5 \]
Carlos, however, incorrectly wrote: \[ 5x = 32.5 \]
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Solving for x: Carlos then divided both sides by 5: \[ x = \frac{32.5}{5} = 6.5 \] But the correct step after properly computing should have been: \[ x = \frac{17.5}{5} = 3.5 \]
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Identifying the Mistake: The mistake was made in the subtraction step when Carlos calculated \(25 - 7.5\). Instead of getting \(17.5\), he incorrectly computed it to be \(32.5\).
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Checking the Solution: Carlos could have checked his solution by substituting \(x = 6.5\) back into the original equation. We can also check the correct solution \(x = 3.5\) afterward:
Substitute \(x = 6.5\): \[ 5(6.5) + 7.5 = 25 \] Calculating the left side: \[ 5(6.5) = 32.5 \implies 32.5 + 7.5 = 40 \] Since \(40 \neq 25\), Carlos’s solution is incorrect.
Now let's check the correct solution \(x = 3.5\): \[ 5(3.5) + 7.5 = 25 \] Calculating the left side: \[ 5(3.5) = 17.5 \implies 17.5 + 7.5 = 25 \] Since \(25 = 25\), the correct solution is confirmed.
Conclusion: Carlos's mistake was in the subtraction step. He could have confirmed correctness by substituting his found value back into the original equation, and should have gotten \(40\) instead of \(25\), indicating an error had occurred. The correct answer to the equation is \(x = 3.5\).