Let's go through Jerome's work step-by-step to identify any errors:
1. Equation given:
[tex]\[ 500 + 375x = y \][/tex]
Jerome correctly starts with the given equation that models his savings situation.
2. Addition property of equality:
[tex]\[ 500 + 375x - 500 = y - 500 \][/tex]
Jerome applies the addition property of equality to isolate the term containing [tex]\( x \)[/tex] on one side of the equation.
3. Simplification:
[tex]\[ 375x = y - 500 \][/tex]
Jerome simplifies both sides of the equation correctly after subtracting 500 from both sides.
4. Division property of equality:
[tex]\[ \frac{375z}{375} = \frac{y - 500}{375} \][/tex]
This step contains an error. Jerome mistakenly uses [tex]\( z \)[/tex] instead of [tex]\( x \)[/tex]. It should be:
[tex]\[ \frac{375x}{375} = \frac{y - 500}{375} \][/tex]
5. Simplification:
[tex]\[ x = \frac{y - 500}{375} \][/tex]
After correcting the variable, this step correctly simplifies the equation to solve for [tex]\( x \)[/tex].
6. Substitution, [tex]\( y = 3500 \)[/tex]:
[tex]\[ x = \frac{3500 - 500}{375} \][/tex]
Jerome substitutes 3500 for [tex]\( y \)[/tex], which is correct.
7. Simplification:
[tex]\[ x = \frac{3000}{375} = 8 \][/tex]
Jerome simplifies the fraction correctly to find that [tex]\( x \)[/tex] equals 8.
In summary, the error in Jerome's solution is in Step 4 where he uses [tex]\( z \)[/tex] instead of [tex]\( x \)[/tex]. This discrepancy is the mistake in his process. Hence, the correct error is in Step 4, and the corrected step should be:
[tex]\[ \frac{375x}{375} = \frac{y - 500}{375} \][/tex]
This ensures the variable [tex]\( x \)[/tex] is consistently used throughout the equation.
