Hello everyone. I am stuck on a work question that we are required to solve using the matrix (or Gauss-Jordan) method. "A car rental company wants to buy 100 new cars. Compact cars cost $12,000 each, intermediate size cars cost $18,000 each, full size cars cost $24,000 each, and the company has a budget of $1,500,000. If they purchase twice as many compact cars as intermediate sized cars, determine the number of cars of each type that they buy, assuming they spend the entire budget." I am fairly certain that I could solve this easily, except I cannot figure out the proper three equations that correspond to this question. I someone could help me figure them out, it would be greatly appreciated!

Let c be the cost of compact cars, i be the cost of intermediate cards, and f be the cost of full-size cars. We have the following equations: c + i + f = 100 12,000c + 18,000i + 24,000 f = 1,500,000 c = 2i