What’s the answer? At a concession stand, seven hot dogs and four hamburgers cost $18.50; four hot dogs and seven hamburgers cost $20.00. Find the cost of one hot dog and the cost of one hamburger.

7d+4h = 18.50

4d+7h = 20.00
add and you have
11d+11h = 38.50
1d+1h = 3.50
now use that in either equation to find d and h sepaately

To find the cost of one hot dog and one hamburger, let's set up a system of equations based on the given information.

Let's assume the cost of one hot dog is represented by 'x' dollars and the cost of one hamburger is represented by 'y' dollars.

From the first statement, "seven hot dogs and four hamburgers cost $18.50," we can write the equation:

7x + 4y = 18.50

From the second statement, "four hot dogs and seven hamburgers cost $20.00," we can write the equation:

4x + 7y = 20.00

Now, we have a system of equations:

7x + 4y = 18.50
4x + 7y = 20.00

There are multiple ways to solve this system of equations, such as substitution, elimination, or matrix methods. Let's solve it using the substitution method:

1. Solve the first equation for x:
Subtract 4y from both sides:
7x = 18.50 - 4y
Divide both sides by 7:
x = (18.50 - 4y)/7

2. Substitute the value of x in the second equation:
4((18.50 - 4y)/7) + 7y = 20.00

Now, we can simplify and solve for y:

(74 - 16y)/7 + 7y = 20.00
Multiply through by 7 to get rid of the denominator:
74 - 16y + 49y = 140
Combine like terms:
-16y + 49y = 140 - 74
33y = 66
Divide both sides by 33:
y = 2

Now that we have the value of y (the cost of one hamburger), we can substitute it back into either of the original equations to find x (the cost of one hot dog). Let's use the first equation:

7x + 4(2) = 18.50
7x + 8 = 18.50
Subtract 8 from both sides:
7x = 10.50
Divide both sides by 7:
x = 1.50

Therefore, the cost of one hot dog is $1.50, and the cost of one hamburger is $2.00.