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Jerome's teacher gave him a homework assignment on solving equations. Since he's been thinking about saving for a used car, he decided to use the assignment as an opportunity to model a savings plan.

He already has \[tex]$500, and he plans to save \$[/tex]375 every month. To model the situation, he created this equation, where [tex]\(y\)[/tex] represents the total amount saved for the car and [tex]\(x\)[/tex] represents the number of months since he started saving:

[tex]\[500 + 375x = y\][/tex]

He then solved the equation to determine how many months he would need to save to have enough to purchase the car. Review his work, and seek the error.

\begin{tabular}{|c|c|}
\hline
Justification & Work \\
\hline
1. Given & [tex]\(500 + 375x = y\)[/tex] \\
2. Addition property of equality & [tex]\(500 + 375x - 500 = y - 500\)[/tex] \\
3. Simplification & [tex]\(375x = y - 500\)[/tex] \\
4. Division property of equality & [tex]\(\frac{375x}{375} = \frac{y - 500}{375}\)[/tex] \\
5. Simplification & [tex]\(x = \frac{y - 500}{375}\)[/tex] \\
6. Substitution, [tex]\(y = 3,500\)[/tex] & [tex]\(x = \frac{3,500 - 500}{375}\)[/tex] \\
7. Simplification & [tex]\(x = 8\)[/tex] \\
\hline
\end{tabular}



Answer :

GinnyAnswer
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.

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