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Peter is saving for a down payment of [tex]$50,000[/tex] for a new home. Yesterday, he created an equation that models his current savings plan to determine how long it will take him to reach his goal. In his model, [tex]y[/tex] represents the total amount saved and [tex]x[/tex] represents the number of months since yesterday. Which statement is true?

[tex]\[
\begin{array}{|lrl|}
\hline
\text{Step 1:} & y & = 30,000 + 2,000x \\
\text{Step 2:} & y - 30,000 & = 30,000 + 2,000x - 30,000 \\
\text{Step 3:} & y - 30,000 & = 2,000x \\
\text{Step 4:} & \frac{y - 30,000}{2,000} & = \frac{2,000x}{2,000} \\
\text{Step 5:} & \frac{y - 30,000}{2,000} & = x \\
\text{Step 6:} & \frac{50,000 - 30,000}{2,000} & = x \\
\text{Step 7:} & 10 & = x \\
\hline
\end{array}
\][/tex]

A. Peter used the division property of equality in step 4.
B. Peter used the substitution property in step 5.
C. Peter used the subtraction property of equality in step 5.
D. Peter used the associative property in step 6.

Sagot :

GinnyAnswer
To solve the given problem, let's analyze each step in Peter's procedure and identify the appropriate mathematical properties he used.

Step-by-Step Solution:
1. Step 1: [tex]\( y = 30,000 + 2,000x \)[/tex]

Peter starts with an equation that models his savings plan where [tex]\( y \)[/tex] represents the total savings and [tex]\( x \)[/tex] represents the number of months. He currently has \[tex]$30,000 saved and saves an additional \$[/tex]2,000 per month.

2. Step 2: [tex]\( y - 30,000 = 30,000 + 2,000x - 30,000 \)[/tex]

Peter subtracts \$30,000 from both sides of the equation. This is applying the subtraction property of equality.

3. Step 3: [tex]\( y - 30,000 = 2,000x \)[/tex]

By simplifying the expression [tex]\( 30,000 + 2,000x - 30,000 \)[/tex] on the right side, Peter shows that [tex]\( y - 30,000 = 2,000x \)[/tex].

4. Step 4: [tex]\( \frac{y - 30,000}{2,000} = \frac{2,000x}{2,000} \)[/tex]

Here, Peter divides both sides of the equation [tex]\( y - 30,000 = 2,000x \)[/tex] by 2,000. This is applying the division property of equality.

5. Step 5: [tex]\( \frac{y - 30,000}{2,000} = x \)[/tex]

Simplifying the right side of the equation [tex]\( \frac{2,000x}{2,000} \)[/tex] to get [tex]\( x \)[/tex], Peter shows the equation in terms of [tex]\( x \)[/tex].

6. Step 6: [tex]\( \frac{50,000 - 30,000}{2,000} = x \)[/tex]

Peter substitutes [tex]\( y = 50,000 \)[/tex] (his savings goal) into the equation to determine the number of months [tex]\( x \)[/tex] required to reach this goal. This uses the substitution property.

7. Step 7: [tex]\( 10 = x \)[/tex]

After performing the arithmetic [tex]\( \frac{20,000}{2,000} = 10 \)[/tex], Peter finds that it will take 10 months to reach his savings goal.

Correct Answer:
A. Peter used the division property of equality in step 4.

So, the answer is:
[tex]\( (10.0, 'A') \)[/tex]

Thus, Peter used the division property of equality in step 4.
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