Question

One hundred liters of a 50% solution of a chemical mixture is obtained by mixing a 60% solution with a 20% solution. Using a system of linear equations determine how many liters of each solution are required to obtain the 50% mixture. Solve the system using matrices.

Answer 1

By setting up a system of linear equations and solving using matrices, we find that to create a 50% solution, 60 liters of the 60% solution and 40 liters of the 20% solution are needed.

Step-by-step explanation:

To create a 50% chemical mixture from a 60% and a 20% solution, we can set up a system of linear equations. Let the amount of the 60% solution be x liters and the amount of the 20% solution be y liters. Our first equation comes from the total volume:

x + y = 100

The second equation comes from the percentage of the chemical in the final mixture:

0.60x + 0.20y = 0.50 × 100

We can represent this system as a matrix equation A⋅X = B, where A is the coefficient matrix, X is the variable matrix, and B is the outcome matrix:

A = 1 1 0.60 0.20 ⋅ X = x y B = 100 50

After solving the matrix equation, we find that x = 60 liters and y = 40 liters.