Regina's teacher recently gave her a homework assignment on solving equations. Since she has been thinking about saving for a new cell phone, she decided to use the assignment as an opportunity to model a savings plan.

She created this equation to model the situation. In it, [tex]$y$[/tex] represents the total amount saved for the new cell phone, 74 is the amount of money she has now, 40 is the amount of money she saves each month for the phone, and [tex]$x$[/tex] represents the number of months since she started saving:
[tex]\[
74 + 40x = y
\][/tex]

She then solved the equation to determine how many months she'd need to save to have enough to purchase the new cell phone. Review her work and select the error.

[tex]\[
\begin{array}{|c|c|c|}
\hline
Justification & Steps & Work \\
\hline
\begin{tabular}{l}
1: Given \\
2: Subtraction Property of Equality \\
3: Simplification \\
4: Division Property of Equality \\
5: Simplification \\
6: Substitution, $y = 834$ \\
7: Simplification
\end{tabular}
&
\begin{tabular}{l}
Step 1: \\
Step 2: \\
Step 3: \\
Step 4: \\
Step 5: \\
Step 6: \\
Step 7:
\end{tabular}
&
\begin{tabular}{l}
\begin{aligned}
74 + 40x & = y \\
74 + 40x - 74 & = y - 74 \\
40x & = y - 74 \\
\frac{40x}{40} & = \frac{y - 74}{40} \\
x & = \frac{y - 74}{40} \\
x & = \frac{834 - 74}{40} \\
x & = 19
\end{aligned}
\end{tabular} \\
\hline
\end{array}
\][/tex]

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.