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Regina has $\$ 500$ in a savings account. She wants to purchase a washer and dryer set for $\$ 1,600$, and she takes out an interest-free loan at the store. Her monthly payment is $\$ 134$. Regina saves an extra $\$ 75$ 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

$\begin{array}{l}

y=134 x+500 \\

y=1,600-100 x

\end{array}$


B

$\begin{array}{l}

y=75 x+500 \\

y=1,600-134 x

\end{array}$


C

$\begin{array}{l}

y=500 x+75 \\

y=1,600-134 x

\end{array}$


D

$\begin{array}{l}

y=1,600 x+134 \\

y=75 x-500

\end{array}$

Verified Answer :

  • Regina's total savings increase over time, modeled by \(y = 75x + 500\), where \(75\) is the monthly savings and \(500\) is the initial balance.
  • The remaining loan balance for the \(\$1,600\) set decreases as she makes monthly payments of \(\$134\), modeled by \(y = 1,600 - 134x\).
  • The system that represents this financial situation is \(\boxed{B}\) because it accurately captures both her growing savings and her reducing debt.

Explanation

  1. Modeling Regina's Total Savings

Regina starts with \(\$500\) in her savings account. She saves an additional \(\$75\) each month. If \(x\) represents the number of months that have passed, her total savings \(y\) can be modeled by the linear equation:
\[y = 75x + 500\]
This equation accounts for her starting balance and her recurring monthly savings.

  1. Modeling the Remaining Loan Balance

Regina needs to pay for a washer and dryer set costing \(\$1,600\). She is paying off this set with a loan, and she makes a monthly payment of \(\$134\). As time progresses, the amount she still owes (the remaining balance \(y\)) decreases. This can be modeled by the linear equation:
\[y = 1,600 - 134x\]
This equation shows the total cost starting at \(\$1,600\) and reducing by \(\$134\) for every month \(x\) that passes.

  1. Identifying the Correct System of Equations

To determine when she will have enough money to completely pay for the set, we can analyze the two equations together. The time \(x\) when she can afford the set occurs when her savings balance meets the remaining loan balance. Thus, the system of equations is:
\[ \begin{array}{l} y = 75x + 500 \ y = 1,600 - 134x \end{array} \]
Comparing this result to the given options, we see that it matches Option B.

Examples

Systems of equations are incredibly useful for personal finance and planning. For example, if you are saving up for a new laptop while simultaneously paying off a small debt, you can use a system of equations to determine the exact month when your total savings will exceed your remaining debt. This helps in setting clear financial goals and tracking progress, allowing you to manage your budget effectively and make informed decisions about your purchases.

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