Let's carefully analyze each step of Jerome's solution to identify where an error might have occurred.
Step 1: Subtraction Property of Equality
Jerome started with the equation:
[tex]\[ \frac{1}{3} x + \frac{5}{6} = 1 \][/tex]
He then subtracted [tex]\(\frac{5}{6}\)[/tex] from both sides:
[tex]\[ \frac{1}{3} x + \frac{5}{6} - \frac{5}{6} = 1 - \frac{5}{6} \][/tex]
This simplifies to:
[tex]\[ \frac{1}{3} x = 1 - \frac{5}{6} \][/tex]
Calculating the right side:
[tex]\[ 1 - \frac{5}{6} = \frac{6}{6} - \frac{5}{6} = \frac{1}{6} \][/tex]
So far, there's no error, and we have:
[tex]\[ \frac{1}{3} x = \frac{1}{6} \][/tex]
Step 2: Evaluating the Expression
Jerome mentioned correcting this step to:
[tex]\[ \frac{1}{3} x = \frac{1}{6} - \frac{5}{6} \][/tex]
It appears he tried calculating [tex]\(\frac{1}{6} - \frac{5}{6}\)[/tex]:
[tex]\[ \frac{1}{6} - \frac{5}{6} = \frac{1 - 5}{6} = \frac{-4}{6} = \frac{-2}{3} \][/tex]
At this point, the equation should be:
[tex]\[ \frac{1}{3} x = \frac{-2}{3} \][/tex]
Step 3: Multiply by the Reciprocal
Then, to isolate [tex]\(x\)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{1}{3}\)[/tex], which is [tex]\(3\)[/tex]:
[tex]\[ x = \frac{-2}{3} \times 3 \][/tex]
Simplifying:
[tex]\[ x = -2 \][/tex]
Step 4: The Final Answer
So, the correct value of [tex]\(x\)[/tex] is:
[tex]\[ x = -2 \][/tex]
Conclusion:
Jerome made an error in step 1 when simplifying [tex]\(\frac{1}{3} x = 1 - \frac{5}{6}\)[/tex] by stating that it equals [tex]\(\frac{1}{6}\)[/tex] instead of evaluating it directly. Correctly evaluated, [tex]\(1 - \frac{5}{6} = \frac{1}{6}\)[/tex].
Therefore:
- Jerome made no error in steps 2, 3, and 4.
- The correct calculation would yield [tex]\(x = -2\)[/tex], and any errors in steps 2, 3, and 4 would have caused incorrect results if they were significant.
Thus, despite the arduous presentation and potential confusions, the errors and corrections align - Jerome should ensure correctly simplifying fractional expressions meticulously in all steps consistently.
