SOLUTION: A movie theater charges $7 for adults, $4.50 for children, and $1 for senior citizens. On one day, the theater sold 635 tickets and collected $2910.00 in receipts. there were 2 t

Algebra ->  Matrices-and-determiminant -> SOLUTION: A movie theater charges $7 for adults, $4.50 for children, and $1 for senior citizens. On one day, the theater sold 635 tickets and collected $2910.00 in receipts. there were 2 t      Log On


   

Question 317782: A movie theater charges $7 for adults, $4.50 for children, and $1 for senior citizens. On one day, the theater sold 635 tickets and collected $2910.00 in receipts. there were 2 times as many children's tickets sold as adult tickets. How many adults, children, and senior citizens went to the movies that day? (solve using matrices. find the reduced row echelon form).
x = adults y = children and z = senior citizens
equation 1: x + y + z = 635
equation 2: $7x + $4.5y + $1z = $2910.00
equation 3: x = 2y (IS THIS RIGHT????)
equation 4: x greater than or equal to 0
equation 5: y is greater than or equal to 0
How do I figure out the third equation and solve using a matrix and putting it into reduced row echelon form? I need to know how to figure that part out, I can get it into reduced row echelon form. Your help is appreciated. Thank you!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Since "there were 2 times as many children's tickets sold as adult tickets", this means that y=2x. Let's say that there were x=10 adult tickets. This then means that there are y=2(10)=20 children's tickets since "there were 2 times as many children's tickets sold as adult tickets".


So the third equation is y=2x. If you get everything to one side, you get -2x%2By%2B0z=0

So the three equations are

system%28x%2By%2Bz=635%2C70x%2B45y%2B10z=29100%2C-2x%2By%2B0z=0%29


Note: I multiplied ALL of the terms of the second equation by 10 to make every number a whole number.


which translates into the matrix equation




Now append the right side to the 3x3 matrix on the left side to get the augmented matrix

%28matrix%283%2C4%2C1%2C1%2C1%2C635%2C70%2C45%2C10%2C29100%2C-2%2C1%2C0%2C0%29%29


Now let's row reduce. Note: Solution provided by the Linear Algebra Toolkit




Now look at the last column of the last matrix (ie the last step shown). This column has the values 175, 350, and 110. This means that x=175, y=350 and z=110


So 175 adults, 350 children, and 110 seniors went to the movies that day.