Question

Computing a Refund The grocery store we use does not mark prices on its goods. My wife went to this store, bought three 1-pound packages of bacon and two cartons of eggs, and paid a total of \(\$ 13.45 .\) Not knowing that she went to the store, I also went to the same store, purchased two 1-pound packages of bacon and three cartons of eggs, and paid a total of \(\$ 11.45 .\) Now we want to return two 1-pound packages of bacon and two cartons of eggs. How much will be refunded?

Short answer

The refund will be $9.96.

Step-by-step solution

Formulate the Equations

Let the price of one 1-pound package of bacon be represented by \(x\) and the price of one carton of eggs be represented by \(y\) . According to the problem, two separate purchases were made. The first equation based on the first purchase is: \[3x + 2y = 13.45\]The second equation based on the second purchase is: \[2x + 3y = 11.45\]

Solve the System of Equations

To find the values of \(x\) and \(y\), we need to solve the system of linear equations.Multiply the first equation by 2 and the second equation by 3 to align the coefficients of \(x\) and \(y\):\[6x + 4y = 26.90\]\[6x + 9y = 34.35\]Subtract the first modified equation from the second one to eliminate \(x\):\[(6x + 9y) - (6x + 4y) = 34.35 - 26.90\]\[5y = 7.45\]\[y = 1.49\]

Find the Value of x

Substitute \(y = 1.49\) into one of the original equations to find \(x\). Using the first equation:\[3x + 2(1.49) = 13.45\]\[3x + 2.98 = 13.45\]\[3x = 13.45 - 2.98\]\[3x = 10.47\]\[x = 3.49\]

Calculate the Refund

We need to return two 1-pound packages of bacon and two cartons of eggs. Use the values for \(x\) and \(y\) to calculate the refund amount:\[2x + 2y = 2(3.49) + 2(1.49)\]\[= 6.98 + 2.98\]\[= 9.96\]

Key concepts

linear equations

Linear equations are mathematical expressions that represent a relationship between variables. They are called 'linear' because their graphs form straight lines. In this problem, linear equations help us represent the total cost of the items purchased. We use the variables \( x \) and \( y \) to represent the price of bacon and eggs, respectively. The equations we get from the problem are:
1. \[3x + 2y = 13.45\]
2. \[2x + 3y = 11.45\]
These equations summarize the total amounts paid during the two shopping trips. Then, we solve these equations to find the exact prices. Understanding linear equations is key in solving many real-world problems where relationships between quantities are straightforward.

substitution method

The substitution method is one way to solve systems of linear equations. This method involves solving one of the equations for one variable and then substituting that expression into the other equation. In this problem, suppose we had solved for \( x \) in terms of \( y \) from one of the equations (which we did not here, but it is useful to understand):
1. From \[2x + 3y = 11.45\], solve for \( x \): \[x = \frac{11.45 - 3y}{2}\].
Then substitute this into the other equation \[3x + 2y = 13.45\] to get an equation in terms of \( y \) only. Solving, you would then find \( y \). Once \( y \) is known, you would substitute this value back to find \( x \). While we used elimination in our solution, the substitution method is handy as an alternative approach. It sometimes simplifies the equations more easily in certain scenarios.

elimination method

The elimination method simplifies solving systems of equations by eliminating one of the variables. This is done by adding or subtracting the equations after aligning their coefficients. In this problem:
1. Multiply the first equation by 2: \[6x + 4y = 26.90\].
2. Multiply the second equation by 3: \[6x + 9y = 34.35\].
By subtracting the first modified equation from the second, \[6x + 9y - (6x + 4y) = 34.35 - 26.90\], we eliminate \( x \): \[5y = 7.45\].
Solving for \( y \): \[y = 1.49\].
Substitute \( y \) back into one of the original equations to find \( x \). Here, using elimination made it straightforward to get rid of one variable quickly, allowing us to solve the system efficiently.

refund calculation

Calculating the refund involves using the prices found for the items. After determining \( x = 3.49 \) and \( y = 1.49 \):
1. We need to return 2 packages of bacon and 2 cartons of eggs.
2. Using the equation \[2x + 2y\]:\[2(3.49) + 2(1.49) = 6.98 + 2.98 = 9.96\].
This refund calculation shows us that the total amount to be refunded is \$9.96 \. By breaking down the problem into defining variables, forming equations, solving them, and finally using the values to find the refund, we apply systematic problem-solving methods effectively.