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?

Verified step by step guidance

1. Define the variable: Let $x$ represent the number of ounces of the 50% alcohol solution to be mixed.
2. Set up the equation based on the total amount of alcohol in the mixture: The amount of pure alcohol from the 50% solution is $0.50x$, and from the 20% solution is $0.20\; imes\; 80$.
3. Express the total amount of alcohol in the final mixture: The total volume is $x\; +\; 80$ ounces, and the concentration is 40%, so the amount of alcohol is $0.40(x\; +\; 80)$.
4. Write the equation equating the sum of alcohol amounts to the final alcohol amount: $0.50x\; +\; 0.20\; imes\; 80\; =\; 0.40(x\; +\; 80)$.
5. Solve the equation for $x$ by first expanding and simplifying both sides, then isolating $x$ to find the number of ounces of the 50% solution needed.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. In mixture problems, these equations represent relationships between quantities and concentrations.

Mixture Problems and Concentration

Mixture problems involve combining substances with different concentrations to achieve a desired concentration. The key is to set up equations based on the total amount and the amount of the substance of interest (e.g., alcohol) in each solution before and after mixing.

Setting Up Equations from Word Problems

Translating a word problem into equations requires identifying variables, writing expressions for quantities and concentrations, and forming equations that represent the problem's conditions. Careful interpretation ensures the system accurately models the scenario for solving.