21. Ms. Proctor generated an equivalent expression by simplifying She made a mistake. What was her mistake? $$ \begin{array}{l}\qquad(x+4 y)= \ 9 x+36 y= \ 45 x y\end{array} $$

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UpStudy AI · Accepted Answer

To analyze Ms. Proctor's work, let's break down the expressions step by step.

1. **Starting Expression**: 
   $$
   (x + 4y)
   $$

2. **First Step**: 
   Ms. Proctor seems to have simplified this expression to:
   $$
   9x + 36y
   $$
   This step is incorrect. The expression $$ (x + 4y) $$ cannot be simplified to $$ 9x + 36y $$.

The correct interpretation of $$ (x + 4y) $$ is simply $$ x + 4y $$. There is no multiplication or distribution that leads to $$ 9x + 36y $$.

3. **Second Step**: 
   Then, she equated $$ 9x + 36y $$ to:
   $$
   45xy
   $$
   This is also incorrect. The expression $$ 9x + 36y $$ cannot be equated to $$ 45xy $$ because they are not equivalent expressions.

The expression $$ 45xy $$ represents a product of $$ 45 $$, $$ x $$, and $$ y $$, while $$ 9x + 36y $$ is a sum of two terms.

### Summary of Mistakes:
- The first mistake was incorrectly simplifying $$ (x + 4y) $$ to $$ 9x + 36y $$.
- The second mistake was equating $$ 9x + 36y $$ to $$ 45xy $$, which are not equivalent expressions.

### Conclusion:
Ms. Proctor's mistake lies in both the incorrect simplification of the expression and the incorrect equation of two non-equivalent expressions. The correct expression remains $$ x + 4y $$.
Ms. Proctor's mistake was simplifying $$ (x + 4y) $$ to $$ 9x + 36y $$ and then equating it to $$ 45xy $$. The correct expression is $$ x + 4y $$.

21. Ms. Proctor generated an equivalent expression by simplifying She made a mistake. What was her mistake? \[ \begin{array}{l}\qquad(x+4 y)= \\ 9 x+36 y= \\ 45 x y\end{array} \]

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