Question

Jerry solved this equation: 3 ( x −
1
4
) =
13
6

1. 3x −
3
4
=
13
6

2. 3x −
3
4
+
3
4
=
13
6
+
3
4

3. 3x =
26
12
+
9
12

4. 3x =
35
12

5. (
3
1
)
3
1
x =
35
12
(
3
1
)

6. x =
105
12

In which step did Jerry make an error?
In step 2, he should have subtracted Three-fourths from both sides.
In step 3, he should have found an LCD of 10.
In step 4, he should have subtracted 9 from 26.
In step 5, he should have multiplied both sides by One-third.

Answer 1

Let's analyze Jerry's steps one by one to identify where he made an error:

1. Starting from the original equation:
\[
3 \left( x - \frac{1}{4} \right) = \frac{13}{6}
\]
This step correctly expands to:
\[
3x - \frac{3}{4} = \frac{13}{6}
\]

2. In the second step, Jerry correctly adds \(\frac{3}{4}\) to both sides:
\[
3x - \frac{3}{4} + \frac{3}{4} = \frac{13}{6} + \frac{3}{4}
\]

However, at this step Jerry makes an error in calculating \(\frac{13}{6} + \frac{3}{4}\). The least common denominator (LCD) of \(6\) and \(4\) is \(12\):

- Convert \(\frac{13}{6}\) to have a denominator of \(12\):
\[
\frac{13}{6} = \frac{13 \times 2}{6 \times 2} = \frac{26}{12}
\]

- Convert \(\frac{3}{4}\) to have a denominator of \(12\):
\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\]

Therefore,
\[
\frac{13}{6} + \frac{3}{4} = \frac{26}{12} + \frac{9}{12} = \frac{35}{12}
\]

So, step 2 is correct, but step 3 shows Jerry incorrectly stating that \(3x\) equals a different expression.

3. In step 3, Jerry's conversion is incorrect: \(3x\) does not equal \(26/12 + 9/12\), since he should have computed:
\[
3x = \frac{26}{12} + \frac{9}{12} = \frac{35}{12}
\]

Hence, \(3x = 35/12\) in step 4 is actually stated correctly in that step; however, he should've reached it correctly through the previous steps.

4. Step 4 accurately states:
\[
3x = \frac{35}{12}
\]

5. Step 5 attempts to simplify by multiplying both sides by \(\frac{1}{3}\), which is needed to solve for \(x\).

From our analysis, **the error occurs in step 3** when he did not correctly combine the fractions.

So the correct answer is: **In step 3, he should have found an LCD of 12** (not 10, since 10 is incorrect). Specifically, he should have added \(26/12 + 9/12\) properly.