SOLUTION: A tour group split into two groups when waiting in line for food at a fast food counter. The first group bought 8 slices of pizza and 4 soft drinks for $36.12. The second grou

Algebra ->  Customizable Word Problem Solvers  -> Finance -> SOLUTION: A tour group split into two groups when waiting in line for food at a fast food counter. The first group bought 8 slices of pizza and 4 soft drinks for $36.12. The second grou      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 901886: A tour group split into two groups when waiting in line for food at a fast
food counter.
The first group bought 8 slices of pizza and 4 soft drinks for $36.12.
The second group bought 6 slices of pizza and 6 soft drinks for $31.74.
How much does one slice of pizza cost?
The answer is 3.74 per slice of pizza, but my task is to figure out how to get the answer. I do not understand this at all, so if you could help me I would really appreciate it.
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
8p+4s=36.12
6p+6s=31.74
There are many ways to solve this.
Which way are you learning?
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

8%2Ax%2B4%2Ay=3612
6%2Ax%2B6%2Ay=3174

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 8 and 6 to some equal number, we could try to get them to the LCM.

Since the LCM of 8 and 6 is 24, we need to multiply both sides of the top equation by 3 and multiply both sides of the bottom equation by -4 like this:

3%2A%288%2Ax%2B4%2Ay%29=%283612%29%2A3 Multiply the top equation (both sides) by 3
-4%2A%286%2Ax%2B6%2Ay%29=%283174%29%2A-4 Multiply the bottom equation (both sides) by -4


So after multiplying we get this:
24%2Ax%2B12%2Ay=10836
-24%2Ax-24%2Ay=-12696

Notice how 24 and -24 add to zero (ie 24%2B-24=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2824%2Ax-24%2Ax%29%2B%2812%2Ay-24%2Ay%29=10836-12696

%2824-24%29%2Ax%2B%2812-24%29y=10836-12696

cross%2824%2B-24%29%2Ax%2B%2812-24%29%2Ay=10836-12696 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-12%2Ay=-1860

y=-1860%2F-12 Divide both sides by -12 to solve for y



y=155 Reduce


Now plug this answer into the top equation 8%2Ax%2B4%2Ay=3612 to solve for x

8%2Ax%2B4%28155%29=3612 Plug in y=155


8%2Ax%2B620=3612 Multiply



8%2Ax=3612-620 Subtract 620 from both sides

8%2Ax=2992 Combine the terms on the right side

cross%28%281%2F8%29%288%29%29%2Ax=%282992%29%281%2F8%29 Multiply both sides by 1%2F8. This will cancel out 8 on the left side.


x=374 Multiply the terms on the right side


So our answer is

x=374, y=155

which also looks like

(374, 155)

Notice if we graph the equations (if you need help with graphing, check out this solver)

8%2Ax%2B4%2Ay=3612
6%2Ax%2B6%2Ay=3174

we get



graph of 8%2Ax%2B4%2Ay=3612 (red) 6%2Ax%2B6%2Ay=3174 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (374,155). This verifies our answer.