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.
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.
