Answer :
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.
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.