To determine how many months Regina needs to save to have enough to purchase the new cell phone, we analyze her equation step by step:
### Step-by-Step Solution
Given Equation:
[tex]\[ 74 + 40x = y \][/tex]
1. Given:
[tex]\[ 74 + 40x = y \][/tex]
2. Subtraction Property of Equality:
Subtract 74 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 74 + 40x - 74 = y - 74 \][/tex]
3. Simplification:
[tex]\[ 40x = y - 74 \][/tex]
4. Division Property of Equality:
Divide both sides by 40 to solve for [tex]\( x \)[/tex]:
[tex]\[ \frac{40x}{40} = \frac{y - 74}{40} \][/tex]
5. Simplification:
[tex]\[ x = \frac{y - 74}{40} \][/tex]
6. Substitution:
Substitute [tex]\( y = 834 \)[/tex] (the total amount needed for the cell phone):
[tex]\[ x = \frac{834 - 74}{40} \][/tex]
7. Simplification:
Perform the arithmetic in the numerator first:
[tex]\[ 834 - 74 = 760 \][/tex]
Then divide by 40:
[tex]\[ x = \frac{760}{40} = 19 \][/tex]
So, Regina needs 19 months to save enough money for the new cell phone.
### Identifying the Error in Regina's Work
Regina incorrectly calculated her final step, mistakenly reaching the conclusion of [tex]\( x = 19 \)[/tex] using correct mathematical steps. However, this solves to [tex]\( x = 19 \)[/tex] which is accurate.
[tex]\[ \boxed{19} \][/tex]
Therefore, the final answer of 19 months is correct, confirming that Regina had no miscalculation in her process.
