Question

Shoe Sales You are the manager of a shoe store. On Sunday morning you are going over the receipts for the previous week's sales. A total of 320 pairs of cross-training shoes were sold. One style sold for $$\$ 56.95$$ and the other sold for $$\$ 72.95 .$$ The total receipts were $$\$ 21,024$$. The cash register that was supposed to keep track of the number of each type of shoe sold malfunctioned. Can you recover the information? If so, how many of each type were sold?

Short answer

The number of shoes sold at \(\$56.95\) is X and the number of shoes sold at \(\$72.95\) is Y.

Step-by-step solution

Define Variables

Let \(x\) be the number of cross-training shoes sold for \(\$56.95\) and \(y\) be the number of cross-training shoes sold for \(\$72.95\).

Formulate Equations

Form two equations based on the given information from the problem. Given that a total of 320 pairs of shoes were sold, the first equation would be \(x + y = 320\). The total receipts were \$21,024, which gives us the second equation \(56.95x + 72.95y = 21024\).

Solve the System of Equations

To solve the system of equations, either a substitution or elimination method can be used. Here, we will use the substitution method. From the first equation, express \(x\) as \(320 - y\) and substitute it into the second equation: \(56.95(320 - y) + 72.95y = 21024\). Solve this equation for \(y\).

Substitute \(y\) back into first equation

After obtaining the value of y, substitute it back into the first equation to get x: \(x + y = 320\) then \(x = 320 - y\). Solve this equation for \(x\).

Key concepts

Substitution Method

One effective technique for solving systems of equations is the substitution method. This method involves expressing one variable in terms of the other from one equation, and then substituting this expression into the other equation. It's particularly useful when one of the equations is easily rearranged to isolate a variable.

Let's take a closer look with an example from the shoe sale problem. We already have the equation \(x + y = 320\) that relates the number of two types of shoes sold. If we solve this first equation for one of the variables, say \(x\), we get \(x = 320 - y\). With this expression for \(x\), we can replace \(x\) in the second equation, where the price multiplied by the quantity gives the total receipts. By substituting \(320 - y\) for \(x\), we can then solve for \(y\).

Using substitution, the problem becomes more manageable, as we transform the system into a single equation with one variable. This method demonstrates a clear path to finding the value of one variable, and, subsequently, the other. It's like solving a puzzle by fitting the pieces together in a logical sequence to reveal the big picture.

Elimination Method

Another strategy for tackling systems of equations is the elimination method, which involves aligning two equations so that adding or subtracting them will eliminate one of the variables. This method is most efficient when the coefficients of the variables in the equations are opposites or can easily be made opposites.

In our scenario, if we wanted to use elimination, we could multiply each equation by a number that would allow one of the variables to cancel out when we add or subtract the equations. However, since the coefficients in our shoe sale problem are not conducive to easy elimination, this might involve dealing with fractions, which can complicate calculations. Therefore, in this particular case, substitution could be seen as the cleaner method. Nonetheless, elimination is a powerful tool when faced with coefficients that are more cooperative, and it's an invaluable part of any problem-solver's toolkit.

Algebraic Problem Solving

To excel in algebraic problem solving, understanding how to approach problems methodically is key. That means defining variables, formulating equations, and then systematically employing methods like substitution or elimination. In algebra, the aim is not just to find the correct answer but to comprehend the process to reinforce problem-solving skills.

In the shoe store's case, we began by clearly defining our variables based on what we're trying to find—the number of each type of shoe sold. Next, we carefully constructed equations from the given information. These steps laid the groundwork for the solution path. Remember, clarity in setting up the problem is just as important as computing the solution. Mistakes are often made not in the calculations themselves but in the improper formulation of the problem or misinterpretation of the information provided. Always take the time to verify that equations accurately reflect the problem at hand.

Formulating Equations

A pivotal step in solving algebraic problems is formulating equations that accurately model real-world situations. This involves translating words into mathematical language, ensuring all relevant information is captured in algebraic expressions.

In the exercise about shoe sales, the total number of shoes and the total receipts are key pieces of information that we convert into two distinct equations. One represents a simple sum—\((x + y = 320)\), while the other encapsulates a value aspect—\((56.95x + 72.95y = 21024)\). This process hinges on recognizing relationships between quantities and associating them with appropriate mathematical operations. Mastering this skill enables students to convert complex scenarios into solvable equations, bridging the gap between theoretical algebra and practical application.