An artist can make two kinds of ornaments, free standing or hanging. He can make at most 50 ornaments each week, and the greatest number of hours he can work in a week is 70. It takes 2 hours to make a free standing ornament and 1 hour to make a hanging ornament. His profit is $9 for each free standing ornament and $12 for each hanging ornament. How many of each type ornament should he make to maximize his profit

So the first thing we do is to multiply the top equation by 3. the reason i want to do this is i notice there will be a 6y on both equations and can possibly add/subtract them. so we get 6y = -3x + 27. adding that and the second equation we get 6x = 27-15 which is 12. divide by 6 and we get 2. ∴ x = 2. substituting 2 into the first equation we get 2y = -2 + 9 which is 7. divide by 2 and we get 7/2. ∴ y = 7/2, x = 2.