At a concession stand, three hot dogs and four hamburgers cost $12.00; four hot dogs and three hamburgers cost $10.75. Find the cost of one hot dog and the cost of one hamburger.
To find the cost of one hot dog and the cost of one hamburger, we can set up a system of equations.
Let's assume that the cost of one hot dog is represented by x dollars, and the cost of one hamburger is represented by y dollars.
From the given information, we can form two equations:
Equation 1: 3x + 4y = 12.00 (from the first statement "three hot dogs and four hamburgers cost $12.00")
Equation 2: 4x + 3y = 10.75 (from the second statement "four hot dogs and three hamburgers cost $10.75")
We can solve this system of equations to find the values of x and y.
One way to do this is by using the elimination method. Multiply Equation 1 by 3 and Equation 2 by 4 to eliminate the y term:
Equation 1 (modified): 9x + 12y = 36.00
Equation 2 (modified): 16x + 12y = 43.00
Now, subtract Equation 1 (modified) from Equation 2 (modified) to eliminate the y term:
(16x + 12y) - (9x + 12y) = 43.00 - 36.00
7x = 7.00
x = 7.00 / 7
x = 1.00
Now that we know the value of x, we can substitute it back into either Equation 1 or Equation 2 to find y. Let's use Equation 1:
3x + 4y = 12.00
3(1.00) + 4y = 12.00
3.00 + 4y = 12.00
4y = 12.00 - 3.00
4y = 9.00
y = 9.00 / 4
y = 2.25
Therefore, the cost of one hot dog is $1.00 and the cost of one hamburger is $2.25.
3D+4H=12.00
4D+3H=10.75
D= (3*12-4*10.75)/(det)
H=(3*10.75 -48)/det
where determinate det=9-16