elimanation

Question

While visiting the Putnam County Fair, she first went on the Ferris wheel 3 times and the roller coaster 3 times, using a total of 27 tickets. Then, after taking a break and having a snack, Tess went on the Ferris wheel 4 times and the roller coaster 1 time, using a total of 18 tickets. How many tickets does it take to ride each attraction?

Answer 1

The elimination method adds or subtracts equations to eliminate one of the variables.  Thus, one of the coefficients must be the same, so we often have to multiply one equation (or both) in order to get one of the coefficients to match.

 

Let:  x = number of Ferris wheel tickets per ride

        y = number of roller coaster tickets per ride

 

Note:  the number of tickets is the number of rides times the number of tickets per ride

                    (the number of rides cancels out)

  3x + 3y = 27      (eq1)

  4x +  y  = 18      (eq2)

Get a common coefficient for either x or y.  Let's divide eq1 by 3 and subtract it from eq2.

   4x +  y = 18

    x  +  y  =  9       (modified eq1)

 -----------------

   3x         = 9

    x         = 3        (eq3)       3 tickets/ride for Ferris wheel

    x  + y  = 9        (modified eq1)

 ---------------    (subtract eq3 from modified eq1)

           y  =  6       6 tickets/ride for roller coaster

 

Checking (very important):

  Is  3(3) + 3(6) = 27    ?

         9   +  18   = 27   ?

             27      =  27?  yes

  Is  4(3) + (6)  =  18    ?

        12   +  6   =  18   ?

             18  =   18    ? yes

Answer 2

Let's label our variables. Let f = Ferris wheel and r = roller coaster

Here's what we know:

3f + 3r = 27

4f + r = 18

 

Elimination means that we multiply our variables to the point we can subtract them. If we multiply the second equation by -3, we can remove the r-variable:

 3f  + 3r =  27

-12f - 3r = -54

------------------

-9f = -27

Now we can isolate f by dividing both sides by -9:

f = 3

 

Now that we know f, we can find r:

4 * 3 + r = 18

12 + r = 18

r = 6

 

Now to check by using the other equation:

3 * 3 + 3 * 6 = 27

9 + 18 = 27

27 = 27

Answer 3

x=# tickets for the ferris wheel, y=# tickets for the roller coaster

 

3x+3y=27

4x+y=18

 

Multiply equation 2 by -3:

 

 3x +3y=27

-12x-3y=-54

-------------------

-9x       =-27

         x =3 tickets for the ferris wheel

 

4(3)+y=18

12+y=18

      y=6 tickets for the roller coaster