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Regina has [tex]$\$[/tex]500[tex]$ in a savings account. She wants to purchase a washer and dryer set for $[/tex]\[tex]$1,600$[/tex], and she takes out an interest-free loan at the store. Her monthly payment is [tex]$\$[/tex]134[tex]$. Regina saves an extra $[/tex]\[tex]$75$[/tex] each month in addition to her loan payments. What system of equations could Regina use to determine when she will have enough money to completely pay for the washer and dryer set?

A. [tex]\( y = 134x + 500 \)[/tex]
[tex]\( y = 1,600 - 100x \)[/tex]

B. [tex]\( y = 75x + 500 \)[/tex]
[tex]\( y = 1,600 - 134x \)[/tex]

C. [tex]\( y = 500x + 75 \)[/tex]
[tex]\( y = 1,600 - 134x \)[/tex]

D.
[tex]\[
\begin{aligned}
y &= 1,600x + 134 \\
y &= 75x - 500
\end{aligned}
\][/tex]



Answer :

GinnyAnswer
Let's break down the problem step-by-step to find the appropriate system of equations that Regina could use.

1. Initial Savings and Additional Monthly Savings:
- Regina starts with \[tex]$500 in her savings account. - She saves an additional \$[/tex]75 each month.

We can create an equation to represent the total amount of savings [tex]\(y\)[/tex] Regina will have after [tex]\(x\)[/tex] months. This will include her initial savings plus the additional monthly savings:

[tex]\[ y = 75x + 500 \][/tex]

2. Loan Amount and Monthly Payments:
- Regina takes out a loan of \[tex]$1,600 for the washer and dryer set. - She makes monthly payments of \$[/tex]134.

We can create another equation to represent the remaining loan amount [tex]\(y\)[/tex] after [tex]\(x\)[/tex] months. This will include the initial loan amount minus the total monthly payments made:

[tex]\[ y = 1600 - 134x \][/tex]

Now we need to match these equations with the given options:

- Option A:
[tex]\[ y = 134x + 500 \][/tex]
[tex]\[ y = 1600 - 100x \][/tex]

- Option B:
[tex]\[ y = 75x + 500 \][/tex]
[tex]\[ y = 1600 - 134x \][/tex]

- Option C:
[tex]\[ y = 500x + 75 \][/tex]
[tex]\[ y = 1600 - 134x \][/tex]

- Option D:
[tex]\[ y = 1600x + 134 \][/tex]
[tex]\[ y = 75x - 500 \][/tex]

By comparing our derived equations [tex]\(y = 75x + 500\)[/tex] and [tex]\(y = 1600 - 134x\)[/tex] with the provided options, we see that the equations match Option B.

Therefore, the correct system of equations that Regina should use is:

[tex]\[ \boxed{y = 75x + 500} \][/tex]

[tex]\[ \boxed{y = 1600 - 134x} \][/tex]

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