at a concession stand....

Question

three hot dogs and four hamburgers cost $13.25; four hot dogs and three hamburgers cost $13.00. Find the cost of one hot dog and the cost of one hamburger. 

Answer 1

the difference between the cost of a hot dog and cost of a hamburger is 25 cents and the hamburger costs more. I know that because when you go from 3 hot dogs and 4 burgers to four hot dogs and 3 burgers, the total cost goes down 25 cents

 

4x + 3(x +.25) = 13

 

4x + 3x = 12.25

 

7x = 12.25

 

x = $1.75 for a hot dog

 

$2.00 for a hamburger

Answer 2

So for this problem lets call hot dogs x and hamburgers y. We know that 3 hot dogs and 4 hamburgers costs $13.25. This can be written as the equation

 

3x+4y=13.25

 

Similarly, we know that 4 hotdogs and 3 hamburgers cost $13.00. This gives us the equation

 

4x+3y=$13.00

 

To find the cost of each hot dog(x) and hamburger(y) we need to solve this system of equations.

Start by puting them together, it helps!

 

3x+4y=13.25

4x+3y=13.00

 

I will demonstrate using the Elimination Method. What I am going to do is multiply one of the equations by a number so that when I add the two equations together I cancel the x's or the y's. For my example I will multiply the top equation by (-4/3)

 

(-4/3)(3x+4y=13.25)

 

Simplified this gives us

 

-4x-(16/3)y=(-53/3)

 

Put the equations together

 

-4x-(16/3)y=(-53/3)

4x+3y=13.00

 

As you can see if we add  the equations the x's will cancel out or become 0. So lets do that. MAKE SURE TO DO YOUR FRACTION PROPERLY IF YOU ARENT USING A CALCULATOR!!!

 

(-7/3)y=(-14/3)

 

We simplify this by multiplying both sides by (-3/7). We do this so that y is by itself.

 

y=(-14/3)(-3/7)=2

y=2

 

Now that we know y=2, plug that value into EITHER equation from the system and solve for x. I will substitute it into the first equation to demonstrate.

 

3x+4(2)=13.25

3x+8=13.25

3x=5.25

x=1.75

 

x and y values are the answers. MAKE SURE YOU CONNECT THE VALUES WITH THE ORIGINAL VARIABLE DEFINITIONS!!! We called hot dogs x so hot dogs cost $1.75.  We called hamburgers y so hamburgers cost $2.00

Answer 3

Let d = the cost of one hot dog   and Let h = the cost of one hamburger

1)    3d + 4h = 13.25

2)    4d + 3h = 13.00

 

Mutliply equation one by 3 and equation two by (-4)

 

   9d +12h =  39.75

-16d -12h = -52.00

           -7d = -12.25

           -7           -7

              d = $1.75

 

Substitute the value of d into equation two

 

4(1.75) + 3h = 13.00

           7  + 3h = 13.00

         -7           =  -7.00

                   3h =   6.00

                   3           3

                    h = $2.00