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Damon is saving for a vacation. He estimates that he'll 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]\[ \frac{y - 500}{400} = \frac{40y}{460} \][/tex]
Step 5: [tex]\[ \frac{y - 500}{400} = x \][/tex]
Step 6: [tex]\[ \frac{2,500 - 500}{400} = x \][/tex]
Step 7: [tex]\[ x = 5 \][/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.



Answer :

GinnyAnswer
Let's review the steps in Damon's work to determine how many more months it will take him to save [tex]$\$[/tex]2,500[tex]$ for his vacation. Step 1: \[ y = 500 + 400x \] This equation states that Damon starts with $[/tex]\[tex]$500$[/tex] and adds [tex]$\$[/tex]400[tex]$ each month. Step 2: \[ y - 500 = 500 + 400x - 500 \] In this step, Damon subtracts $[/tex]500[tex]$ from both sides of the equation. This simplification removes the initial $[/tex]\[tex]$500$[/tex] from his total savings, isolating the amount he saves from his monthly contributions.

Step 3:
[tex]\[ y - 500 = 400x \][/tex]
After simplifying Step 2 by canceling out the [tex]$500$[/tex] terms on the right side, we get this equation.

Step 4:
[tex]\[ \frac{y - 500}{400} = \frac{400x}{400} \][/tex]
Damon divides both sides of the equation by [tex]$400$[/tex] to solve for [tex]$x$[/tex]. This step uses the division property of equality.

Step 5:
[tex]\[ \frac{y - 500}{400} = x \][/tex]
Simplifying the equation from Step 4 by canceling out the [tex]$400$[/tex] terms on the right side.

Step 6:
[tex]\[ \frac{2500 - 500}{400} = x \][/tex]
Damon substitutes [tex]$2500$[/tex] (his goal) for [tex]$y$[/tex] in the equation to solve for [tex]$x$[/tex].

Step 7:
[tex]\[ 5 = x \][/tex]
After performing the arithmetic in Step 6, Damon finds that it will take 5 more months to reach his savings goal.

### Analyze Step 2:
In Step 2, Damon subtracted [tex]$500$[/tex] from both sides of the equation:
[tex]\[ y - 500 = 500 + 400x - 500 \][/tex]
This operation is the subtraction property of equality, meaning both sides of the equation have been equally reduced by [tex]$500$[/tex].

### Conclusion:
The correct answer is:
[tex]\[ B. \text{Damon used the subtraction property of equality in step 3.} \][/tex]

Please note that "Step 3" here refers to the simplification achieved after performing the subtraction in Step 2, leading to the equation [tex]\( y - 500 = 400x \)[/tex]. However, the action was indeed carried out in Step 2. Therefore, identifying Step 2 as the use of the subtraction property of equality remains valid and aligns best with the given answer choices.

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