Call hot dogs x and hamburgers y. 5 hot dogs is represented as 5x, and 2 hamburgers is represented as 2y. The cost of these together is 8.50, so that equation is 5x + 2y = 8.50. The second equation where there are 2 hot dogs is represented as 2x, and 5 hamburgers is represented by 5y, and those together cost 10.75. That equation is 2x + 5y = 10.75. We now have a system of equations...2 equations for 2 unknowns. We will use the elimination method to eliminate one variable to solve for the other. Multiply the first equation by -2 and the second equation by 5, and in this process we will eliminate the x terms. Doing that multiplication, we get the 2 equations -10x-4y= -17 and 10x + 25y = 53.75. The x-terms cancel out, leaving us with (-4y = -17)+(25y = 53.75). Doing that addition, we have 21y = 36.75 and y = 1.75. That tells us that the cost of the hamburger is $1.75 each. Fill that value into the y in another equation to solve for x. 2x + 5(1.75) = 10.75. That simplifies to 2x + 8.75 = 10.75 and 2x=2. x = 1, so a hot dog costs $1.
