Use a system of linear equations to solve Exercises 73–84. How many ounces of a 15% alcohol solution must be mixed with 4 ounces of a 20% alcohol solution to make a 17% alcohol solution?

Verified step by step guidance

1. Step 1: Let's denote the amount of the 15% alcohol solution that we need to mix as 'x'. This is what we're trying to find.
2. Step 2: We know that the total amount of alcohol in the final solution should be 17% of the total volume. The total volume is 'x' (the volume of the 15% solution) plus 4 (the volume of the 20% solution). So, the total amount of alcohol is 0.17 * (x + 4).
3. Step 3: The total amount of alcohol is also the sum of the alcohol in the 15% solution and the alcohol in the 20% solution. This gives us the equation: 0.15x (alcohol from the 15% solution) + 0.20 * 4 (alcohol from the 20% solution) = 0.17 * (x + 4).
4. Step 4: Now, we have a linear equation in one variable, 'x'. We can solve this equation by first expanding the right side, then moving all terms involving 'x' to one side and constants to the other side, and finally dividing by the coefficient of 'x'.
5. Step 5: The solution to this equation will give us the amount of the 15% alcohol solution that we need to mix with the 4 ounces of 20% alcohol solution to get a 17% alcohol solution.

Key Concepts

Linear Equations

Linear equations are mathematical statements that establish a relationship between two variables, typically in the form of 'y = mx + b'. In this context, they are used to model the mixing of solutions, where the total amount and concentration of alcohol must be balanced to achieve a desired outcome.

Concentration and Mixture Problems

Concentration refers to the amount of solute (in this case, alcohol) present in a given volume of solution. Mixture problems involve combining different solutions with known concentrations to achieve a target concentration, requiring the use of equations to represent the total volume and the total amount of solute.

System of Equations

A system of equations consists of two or more equations that share variables. To solve the problem, we set up a system that includes equations for the total volume of the mixture and the total amount of alcohol, allowing us to find the unknown quantity of the 15% solution needed to achieve the desired concentration.