Let's carefully review Jerome’s steps to identify and understand any mistakes in his solution.
### Step-by-Step Analysis:
1. Starting Equation:
[tex]\[
500 + 375x = y
\][/tex]
This represents Jerome's total savings [tex]\(y\)[/tex] after [tex]\(x\)[/tex] months when he starts with \[tex]$500 and saves \$[/tex]375 each month.
2. Subtracting 500 from both sides:
[tex]\[
500 + 375x - 500 = y - 500
\][/tex]
Simplifying this gives:
[tex]\[
375x = y - 500
\][/tex]
3. Dividing both sides by 375:
[tex]\[
\frac{375x}{375} = \frac{y - 500}{375}
\][/tex]
Simplifying this gives:
[tex]\[
x = \frac{y - 500}{375}
\][/tex]
4. Substitution:
Jerome’s aim is to find out [tex]\(x\)[/tex], the number of months, needed to save up a total amount [tex]\(y\)[/tex].
Jerome made an error here. He used [tex]\(y = 3375\)[/tex] instead of the correct [tex]\(y = 3500\)[/tex] for substitution.
5. Correct Substitution with [tex]\(y = 3500\)[/tex]:
The correct substitution should be:
[tex]\[
x = \frac{3500 - 500}{375}
\][/tex]
Simplifying this:
[tex]\[
x = \frac{3000}{375}
\][/tex]
[tex]\[
x = 8
\][/tex]
So, the correct number of months required is [tex]\(8\)[/tex] months.
### Identifying the Incorrect Substitution:
When Jerome mistakenly substituted [tex]\(y = 3375\)[/tex] instead of [tex]\(3500\)[/tex], his calculation would be:
[tex]\[
x = \frac{3375 - 500}{375}
\][/tex]
Simplifying this:
[tex]\[
x = \frac{2875}{375}
\][/tex]
[tex]\[
x \approx 7.67
\][/tex]
This results in approximately [tex]\(7.67\)[/tex] months, which is not a typical whole number representation for [tex]\(x\)[/tex], implying the miscalculation.
### Conclusion:
Jerome's error lies in step 7, where he substituted the value [tex]\(y = 3375\)[/tex] instead of the correct [tex]\(y = 3500\)[/tex]. The corrected steps are provided to achieve the accurate result of [tex]\(x = 8\)[/tex] months. This shows the importance of ensuring the correct values are used in substitution to avoid errors in calculations.
