Select the correct answer.

Peter is saving for a down payment of [tex]$\$[/tex]50,000[tex]$ for a new home. Yesterday, he created an equation that models his current savings plan to determine how long it will take him to reach his goal. In his model, $[/tex]y[tex]$ represents the total amount saved and $[/tex]x$ represents the number of months since yesterday. Which statement is true?
\begin{tabular}{rl} \hline
Step 1: & [tex]$y=30,000+2,000x$[/tex] \\
Step 2: & [tex]$y-30,000=30,000+2,000x-30,000$[/tex] \\
Step 3: & [tex]$y-30,000=2,000x$[/tex] \\
Step 4: & [tex]$\frac{y-30,000}{2,000}=\frac{2,000x}{2,000}$[/tex] \\
Step 5: & [tex]$\frac{y-30,000}{2,000}=x$[/tex] \\
Step 6: & [tex]$\frac{50,000-30,000}{2,000}=x$[/tex] \\
Step 7: & 10 & [tex]$=x$[/tex] \\ \hline
\end{tabular}

A. Peter used the division property of equality in step 4.
B. Peter used the substitution property in step 5.
C. Peter used the subtraction property of equality in step 5.
D. Peter used the associative property in step 6.



Answer :

Let's analyze the solution step by step. We are given a sequence of algebraic manipulations that Peter used to determine the number of months required to save $50,000.

Here’s the breakdown:

1. Step 1: \( y = 30,000 + 2,000x \)
- This represents the equation where \( y \) is the total amount saved and \( x \) is the number of months.

2. Step 2: \( y - 30,000 = 30,000 + 2,000x - 30,000 \)
- Peter subtracts $30,000 from both sides to isolate the term with \( x \) on one side.

3. Step 3: \( y - 30,000 = 2,000x \)
- Simplifying the right-hand side from the operation in step 2.

4. Step 4: \( \frac{y - 30,000}{2,000} = \frac{2,000x}{2,000} \)
- Peter divides both sides of the equation by 2,000 to solve for \( x \).

5. Step 5: \( \frac{y - 30,000}{2,000} = x \)
- Simplifying the division on the right-hand side.

6. Step 6: \( \frac{50,000 - 30,000}{2,000} = x \)
- Peter substitutes $50,000 for \( y \) to find out how many months are needed.

7. Step 7: \( 10 = x \)
- Simplifying the division on the left-hand side to find \( x = 10 \).

Now, let's identify the properties used:

- Step 4: Dividing both sides of the equation by the same non-zero number (2,000) is the division property of equality.
- Step 5: This step just simplifies the fraction; it does not use the substitution or subtraction property because it builds directly on the previous division.
- Step 6: Substituting 50,000 for \( y \) into the equation does use the substitution property, but the answer choices might confuse if not clearly interpreting the steps.
- Step 6 also does not involve associative property, which is mainly about grouping.

Hence, the correct statement related to the operations in these steps would be:

A. Peter used the division property of equality in step 4.

This correctly matches the division of both sides in step 4.