91Ó°ÊÓ

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.

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 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.