At a concession stand, five hot dogs and two hamburgers cost $8.50; two hot dogs and five hamburgers cost $10.75. find the cost of one hot dog and the cost of one hamburger

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.