Let's go through the steps Damon used to solve the equation and determine the month count to reach his savings goal. We need to find the true statement about Damon's steps:
Step 1: [tex]\( y = 500 + 400x \)[/tex]
- Damon starts with an equation that models his savings plan. Here, [tex]\( y \)[/tex] represents the total savings, and [tex]\( x \)[/tex] represents the number of months since he started saving.
Step 2: [tex]\( y - 500 = 500 + 400x - 500 \)[/tex]
- Here, Damon subtracts 500 from both sides of the equation. This step simplifies the equation by isolating [tex]\( 400x \)[/tex] on the right-hand side.
Step 3: [tex]\( y - 500 = 400x \)[/tex]
- This step is a result of the simplification done in Step 2. It shows the equation after removing the common terms from both sides.
Step 4: [tex]\(\frac{y-500}{400}=\frac{400x}{400}\)[/tex]
- Damon divides both sides of the equation by 400 to isolate [tex]\( x \)[/tex].
Step 5: [tex]\(\frac{y-500}{400}=x\)[/tex]
- This step is simply the result of the division in Step 4.
Step 6: [tex]\(\frac{2500-500}{400}=x\)[/tex]
- Damon substitutes [tex]\( y \)[/tex] with 2500, the total amount he needs for his trip.
Step 7: [tex]\( \frac{2000}{400} = x \)[/tex]
- Damon divides 2000 by 400 to solve for [tex]\( x \)[/tex].
Step 7 (continued): [tex]\( 5 = x \)[/tex]
- Finally, Damon concludes that he needs 5 months to reach his savings goal.
Given the statements:
A. Damon used the multiplication property of equality in step 2.
- Not correct, because Damon used the subtraction property here.
B. Damon used the subtraction property of equality in step 3.
- Correct, since Damon subtracted 500 from both sides to isolate [tex]\( 400x \)[/tex].
C. Damon used the division property of equality in step 4.
- Correct, because he divided both sides by 400. However, we should select only one correct statement, and since B is the earlier step, we prioritize B.
D. Damon used the associative property in step 5.
- Not correct, Damon simplified the division result here.
E. Damon used the addition property of equality in step 6.
- Not correct, Damon only substituted the value 2500 for [tex]\( y \)[/tex].
The correct statement is [tex]\( \textbf{B} \)[/tex]. Damon used the subtraction property of equality in step 3.
