SOLUTION: This is a word problem question that has to be solved using cramers rule At a college basketball game, student tickets cost 12 dollars each, adult tickets cost 16 dollars each a

This is a word problem question that has to be solved using cramers rule

At a college basketball game, student tickets cost 12 dollars each, 
adult tickets cost 16 dollars each and childrens tickets cost 7 dollars
each. Four times as many adult tickets as children tickets were sold, 
the total number of childrens tickets and adult tickets was half the 
number of student tickets. The total  number of tickets sales was 33,043
dollars. How many of each type of ticket was sold?

Let x = the number of student tickets
Let y = the number of adult tickets
Let z = the number of children's tickets

>>...Four times as many adult tickets as children tickets were sold,..<< 

y = 4z

>>...the total number of childrens tickets and adult tickets was half
the number of student tickets...<<

z + y = (1/2)S


>>...student tickets cost 12 dollars each, adult tickets cost 16 
dollars each and childrens tickets cost 7 dollars each...<<
>>...The total number of tickets sales was 33,043 dollars...<< 

12x + 16y + 7z = 33043

So you have the system:

y = 4z
z + y = (1/2)S
12x + 16y + 7z = 33043

Now we have to rewrite the system so
it can be solved by Cramer's rule.

Rewrite the first equation as

0x + 1y - 4z = 0

Clear the second of fractions by multiplying
through by 2

2z + 2y = x

then rewrite as

-1x + 2y + 2z = 0

Leave the third one as it is, namely

12x + 16y + 7z = 33043

So the system we have to solve is

 0x +  1y - 4z =     0 
-1x +  2y + 2z =     0
12x + 16y + 7z = 33043

The system  must be lined up like the
above to be solveable by Cramer's rule.

Form 4 determinants, D, Dx, Dy, and Dz

To form D, just write all the coefficients
down left of the equal signs, like this:

    | 0   1  -4| 
D = |-1   2   2| 
    |12  16   7| 

Now we will proceed to construct Dx, Dy
and Dz from D

Notice that D does not contain the column
of constants to the right of the equal sign
in the system, which is:

    0
    0
33043

but Dx, Dy, and Dz will all contain it.
Let's call this column the "column of
constants".

x is the FIRST unknown, so replace only the
FIRST column of D with the column of constants,
and we have

     |    0   1  -4| 
Dx = |    0   2   2| 
     |33043  16   7| 

y is the SECOND unknown, so replace only the
SECOND column of D with the column of constants,
and we have

     | 0      0  -4| 
Dy = |-1      0   2| 
     |12  33043   7|

z is the THIRD unknown, so replace only the
THIRD column of D with the column of constants,
and we have

     | 0   1      0| 
Dz = |-1   2      0| 
     |12  16  33043|

Do you know how to eveluate 3 by 3 determinants?
If you don't repost asking how to eveluate a
3 by 3 determinant and we'll show you how.

I will assume you already know how.

    | 0   1  -4| 
D = |-1   2   2| = 191 
    |12  16   7|

     |    0   1  -4| 
Dx = |    0   2   2| = 330430 
     |33043  16   7|

     | 0      0  -4| 
Dy = |-1      0   2| = 132172 
     |12  33043   7|

     | 0   1      0| 
Dz = |-1   2      0| = 33043 
     |12  16  33043|

Now the formulas for x, y and z are

x = Dx/D, y = Dy/D, and z = Dz/D

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

x = Dx/D = 330430/191 = 1730

y = Dy/D = 132172/191 = 692

z = Dz/D = 33043/192 = 173

Edwin