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Let $x$ represent the amount of bread sold and $y$ represent the amount of donuts sold.



A baker sells both bread and donuts. Each loaf of bread costs $\$5.25$ and each donut costs $\$1.25$. One day, the baker noticed that he sold a total of 24 items and made $\$87.75$.



Select the two equations that model this scenario:


  • $5.25x + 1.75y = 24$
  • $x + y = 24$
  • $5.25x + 1.75y = 87$
  • $x + y = 87.75$
  • $y = 5.25x + 24$

Verified Answer :

  • Let \(x\) be the quantity of bread sold and \(y\) be the quantity of donuts sold.
  • The total items sold condition is modeled by the equation \(x + y = 24\).
  • The total revenue condition is modeled by the equation \(5.25x + 1.25y = 87.75\).
  • The correct equations are \(\boxed{x + y = 24}\) and \(\boxed{5.25x + 1.25y = 87.75}\).

Explanation

  1. Analyzing the Problem and Formulating Equations

To model the scenario, we first identify the variables given: let \(x\) be the quantity of bread loaves sold and \(y\) be the quantity of donuts sold. We are given two conditions based on the information from the baker.

First, the total number of items sold is 24. Since the items sold are loaves of bread and donuts, the sum of these quantities must equal the total number of items sold:
\[x + y = 24\]

Second, the total revenue generated is \(\$87.75\). Each loaf of bread costs \(\$5.25\) and each donut costs \(\$1.25\). Thus, the revenue from bread is \(5.25x\) and the revenue from donuts is \(1.25y\). The sum of these revenues must equal the total revenue earned:
\[5.25x + 1.25y = 87.75\]

These two equations represent the system that models the baker's sales for the day.

  1. Selecting the Correct Equations

We compare our derived equations with the options provided:

  • \(5.25x + 1.25y = 24\) (Incorrect: mixes revenue coefficients with item count)
  • \(x + y = 24\) (Correct: models total items sold)
  • \(5.25x + 1.25y = 87.75\) (Correct: models total revenue)
  • \(x + y = 87.75\) (Incorrect: mixes item count with revenue total)
  • \(y = 5.25x + 24\) (Incorrect: does not correctly represent either condition)

The two equations that model the scenario are \(x + y = 24\) and \(5.25x + 1.25y = 87.75\).

Examples

In real life, systems of linear equations are used constantly to balance budgets or manage inventory. For instance, if you are buying a mixture of two types of stationery for an office, say pens at \(\$2.00\) and notebooks at \(\$5.00\), and you have a total budget of \(\$50.00\) for exactly \(16\) items, you can set up a system of equations: \(x + y = 16\) and \(2x + 5y = 50\). Solving this system helps you determine exactly how many of each item you can afford to buy while meeting your specific constraints.

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