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]
