SOLUTION: Problem: The Jurassic Zoo charges $28.00 for each adult admission and $12.00 for each child. The total bill for 454 people from a school trip was $3860.00. How many adults and how

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Problem: The Jurassic Zoo charges $28.00 for each adult admission and $12.00 for each child. The total bill for 454 people from a school trip was $3860.00. How many adults and how      Log On


   

Question 199611: Problem: The Jurassic Zoo charges $28.00 for each adult admission and $12.00 for each child. The total bill for 454 people from a school trip was $3860.00. How many adults and how many children went to the zoo? how would set this problem up useing polynomials
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
let adults be given by x
let children be given by y
x+%2B+y+=+454
28x+%2B+12y+=+3860
Now solve this set of equations to see if there is an 'reasonable' answer
As you can, the answer ends up being a negative number of adults. So the numbers given in the problem aren't 'correct'
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=454
28%2Ax%2B12%2Ay=3860

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 1 and 28 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 28 is 28, we need to multiply both sides of the top equation by 28 and multiply both sides of the bottom equation by -1 like this:

28%2A%281%2Ax%2B1%2Ay%29=%28454%29%2A28 Multiply the top equation (both sides) by 28
-1%2A%2828%2Ax%2B12%2Ay%29=%283860%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
28%2Ax%2B28%2Ay=12712
-28%2Ax-12%2Ay=-3860

Notice how 28 and -28 add to zero (ie 28%2B-28=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2828%2Ax-28%2Ax%29%2B%2828%2Ay-12%2Ay%29=12712-3860

%2828-28%29%2Ax%2B%2828-12%29y=12712-3860

cross%2828%2B-28%29%2Ax%2B%2828-12%29%2Ay=12712-3860 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:

16%2Ay=8852

y=8852%2F16 Divide both sides by 16 to solve for y



y=2213%2F4 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=454 to solve for x

1%2Ax%2B1%282213%2F4%29=454 Plug in y=2213%2F4


1%2Ax%2B2213%2F4=454 Multiply



1%2Ax%2B2213%2F4=454 Reduce



1%2Ax=454-2213%2F4 Subtract 2213%2F4 from both sides

1%2Ax=1816%2F4-2213%2F4 Make 454 into a fraction with a denominator of 4

1%2Ax=-397%2F4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%28-397%2F4%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=-397%2F4 Multiply the terms on the right side


So our answer is

x=-397%2F4, y=2213%2F4

which also looks like

(-397%2F4, 2213%2F4)

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

1%2Ax%2B1%2Ay=454
28%2Ax%2B12%2Ay=3860

we get



graph of 1%2Ax%2B1%2Ay=454 (red) 28%2Ax%2B12%2Ay=3860 (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 (-397%2F4,2213%2F4). This verifies our answer.