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Damon is saving for a vacation. He estimates that he will need about [tex]\$2{,}500[/tex] for the trip. He created an equation last week to model his savings plan to determine how many more months of saving it will take to reach his goal. In the equation Damon created, [tex]x[/tex] represents the number of months since last week and [tex]y[/tex] represents the total amount he's saved for the vacation. Review his work, and select the true statement.

Step 1: [tex]y = 500 + 400x[/tex]

Step 2: [tex]y - 500 = 500 + 400x - 500[/tex]

Step 3: [tex]y - 500 = 400x[/tex]

Step 4: [tex]\frac{y - 500}{400} = \frac{400x}{400}[/tex]

Step 5: [tex]\frac{y - 500}{400} = x[/tex]

Step 6: [tex]\frac{2{,}500 - 500}{400} = x[/tex]

Step 7: [tex]5 = x[/tex]

A. Damon used the multiplication property of equality in step 2.

B. Damon used the subtraction property of equality in step 3.

C. Damon used the division property of equality in step 4.

D. Damon used the associative property in step 5.

E. Damon used the addition property of equality in step 6.

Sagot :

GinnyAnswer
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.
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