Use a system of linear equations to solve Exercises 73–84. How many ounces of a 50% alcohol solution must be mixed with 80 ounces of a 20% alcohol solution to make a 40% alcohol solution?

Key Concepts

Linear Equations

Linear equations are mathematical statements that express the equality of two linear expressions. In the context of this problem, they are used to represent the relationships between the quantities of different alcohol solutions. Each equation corresponds to a specific condition, such as the total volume of the mixture and the concentration of alcohol.

Concentration and Mixture Problems

Concentration refers to the amount of solute (in this case, alcohol) present in a solution relative to the total volume of the solution. Mixture problems often involve combining different solutions with known concentrations to achieve a desired concentration. Understanding how to set up equations based on the concentrations and volumes of the solutions is crucial for solving this type of problem.

System of Equations

A system of equations consists of two or more equations that share variables. To solve the problem, we need to create a system that includes equations for both the total volume of the mixture and the total amount of alcohol. Solving this system will yield the unknown quantity of the 50% alcohol solution needed to achieve the desired concentration.