Problem 75 - College Algebra

Question

Use a system of linear equations with two variables and two equations to solve. A store clerk sold 60 pairs of sneakers. The high-tops sold for $$\$ 98.99$$ and the low-tops sold for $$\$ 129.99 .$$ If the receipts for the two types of sales totaled $$\$ 6,404.40,$$ how many of each type of sneaker were sold?

Step-by-Step Solution

Answer

45 high-tops and 15 low-tops were sold.

Step 1: Define the Variables

Let \( h \) represent the number of high-tops sold, and \( l \) represent the number of low-tops sold.

Step 2: Set Up the System of Equations

From the problem, we know two things: the total number of sneakers sold is 60, and the total revenue is $6,404.40. We can create the following system of equations based on this information:1. \( h + l = 60 \)2. \( 98.99h + 129.99l = 6404.40 \)

Step 3: Solve the First Equation for One Variable

Solve the first equation for \( h \):\[ h = 60 - l \]

Step 4: Substitute into the Second Equation

Substitute \( h = 60 - l \) into the second equation:\[ 98.99(60 - l) + 129.99l = 6404.40 \]

Step 5: Simplify and Solve the Equation

Distribute and simplify the equation:\[ 5939.40 - 98.99l + 129.99l = 6404.40 \]Combine like terms:\[ 5939.40 + 31l = 6404.40 \]Subtract 5939.40 from both sides:\[ 31l = 465 \]Divide by 31:\[ l = 15 \]

Step 6: Solve for the Other Variable

Substitute \( l = 15 \) back into the equation \( h = 60 - l \):\[ h = 60 - 15 = 45 \]

Step 7: Verify the Solution

To ensure the solution is correct, check both the number of sneakers and total revenue:- Total sneakers: \( 45 + 15 = 60 \), which matches the problem statement.- Total revenue: \( 98.99 \times 45 + 129.99 \times 15 = 6404.40 \), matching the given total sales.This confirms the solution is correct.

Key Concepts

Two Variables

When dealing with systems of linear equations, two variables play a crucial role. In this particular problem, the two variables are the different types of sneakers sold, high-tops and low-tops. We denote these variables as \( h \) for high-tops and \( l \) for low-tops. The problem tells us about two different quantities involving these variables: the total number of sneakers sold and the total amount of money earned from the sales.

By representing the high-tops and low-tops with variables, we can simplify the real-world situation into two mathematical equations. This allows us to solve these equations systematically.

• \( h \) = number of high-tops sold
• \( l \) = number of low-tops sold

Understanding what each variable represents in the context of the problem is the first step toward generating a solution.

Linear Equations Solution

Solving a system of linear equations involves finding the values of the variables that satisfy both equations simultaneously. In our sneaker problem, we have two linear equations:

• \( h + l = 60 \) - this represents the total number of sneakers sold.
• \( 98.99h + 129.99l = 6404.40 \) - this represents the total sales revenue.

The typical approach is to solve one equation for one variable and then substitute it into the other equation. This method is known as substitution. We solve the first equation for \( h \) as \( h = 60 - l \). Substituting this into the second equation allows us to find the value of \( l \). Once we have \( l \), we substitute it back to find \( h \). This systematic process ensures we find values that satisfy both equations, thus solving the linear equations.

Algebraic Problem-Solving

Algebraic problem-solving is a powerful tool that helps simplify and systematically address problems like our sneaker example. The key steps are as follows:

• Define the variables: Start by deciding what each variable will represent. This step translates the real-world situation into mathematical terms.
• Set up equations: Use the given information to write equations that express relationships between the variables.
• Choose a method to solve: In most cases, either substitution or elimination is used to find the solution.

Each step builds on the one before it, creating a logical path to the solution. Algebra allows us to deal with complex problems by breaking them down into manageable parts, transforming them into equations, and then methodically solving those equations. This structured approach ensures clarity and precision in problem-solving.