Carlos solved the equation below for x using inverse operations. His solution is x = 6.5. What mistake did Carlos make and how could he have checked his solution to know for certain that his answer is correct?

Question

5x + 7.5 = 25
5x = 32.5
x = 6.5
How could Carlos have checked his solution? Show each step by plugging his solution for x back into the equation.

Answer 1

Let's analyze Carlos's solution step-by-step:

1. Starting Equation: $$5x+7.5=25$$

2. 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: $$5x=32.5$$

3. 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: $$x=\frac{17.5}{5}=3.5$$

4. 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 $32.5$.

5. 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: $$5(6.5)=32.5⟹32.5+7.5=40$$ Since $40\ne 25$, 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: $$5(3.5)=17.5⟹17.5+7.5=25$$ 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 $x=3.5$.