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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 :

  • Define the savings equation as the initial amount plus monthly savings: \(y = 75x + 500\).
  • Define the debt reduction equation as the total cost minus monthly payments: \(y = 1,600 - 134x\).
  • Match these expressions to the provided options.
  • Conclude that option b is the correct system: \(\boxed{b}\).

Explanation

  1. Defining the Equations for Savings and Debt

To model Regina's situation, we define \(x\) as the number of months and \(y\) as the monetary balance. We have two separate financial components: her savings account and the debt remaining on the appliance set.

  • First, for her savings: She starts with an initial balance of \(\$500\) and adds an additional \(\$75\) each month. This is represented by the linear equation \(y = 75x + 500\).
  • Second, for the loan balance: The total cost of the washer and dryer set is \(\$1,600\), and she makes a monthly payment of \(\$134\). The remaining balance on her debt after \(x\) months is represented by the linear equation \(y = 1,600 - 134x\).
  1. Evaluating the Given Options

We now look at the given options to see which system correctly matches our derived equations:

  • Option (a): \(y = 134x + 500\) and \(y = 1,600 - 100x\). This does not match our derived equations.
  • Option (b): \(y = 75x + 500\) and \(y = 1,600 - 134x\). This matches both the savings model and the debt model perfectly.
  • Option (c): \(y = 500x + 75\) and \(y = 1,600 - 134x\). The first equation has incorrect coefficients.
  • Option (d): \(y = 1,600x + 134\) and \(y = 75x - 500\). Neither equation reflects the problem constraints.

Therefore, the system of equations in option (b) is the correct model for determining when Regina will have paid off the debt.

Examples

Systems of linear equations are widely used in financial planning, such as comparing the growth of savings against the reduction of debt. For example, a student might use a similar system to track how quickly their summer job earnings (savings) will accumulate to cover the cost of a new laptop, while also accounting for any installments or recurring expenses. This type of modeling helps individuals make informed decisions about their budget and financial goals.

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