Question 73828
Write an equation for each statement; a=adults, b=kids, c=old folks
:
"A total of 434 people attended a community theatre performance."
:
eq1: a + b + c = 434
:
"The admission prices were $9.00 for adults, $7.50 for students, and $8.00 for senior citizens.  The ticket sales totaled $3712."
:
eq2: 9a + 7.5b + 8c = 3712
:
"At the theatre's next board meeting, the finance manager proposed that for the next play they raise prices to $11.00 for adults, $8.50 for students, and $8.50 for senior citizens. Ticket sales would be 4374"
:
eq3: 11a + 8.5b + 8.5c = 4374
:
He said that if the exact same number of people attend the next performance, the ticket sales would be $4374.  How many adults, students, and senior citizens attended the play?
:
A matrix could be used it would be:
1 +  1 +  1 +  434
9 + 7.5 + 8 + 3712
11 + 8.5 + 8.5 + 4374
:
However elimination can also be used, here is that method.
:
Subtract eq 2 from eq 3
Then subtract eq 1 from that result:
:
11a + 8.5b + 8.5c =4374
 9a + 7.5b +  8c = 3712
------------------------ subtract:
2a + 1b + .5c = 662
1a + 1b + 1c =  434
---------------------- subtracting eliminates b:
1a + 0b - .5c = 228
a - .5c = 228; a two unknown equation
:
Multiply equation 1 by 7.5 and subtract it from equation 2:
 9a  + 7.5b +  8c  = 3712
7.5a + 7.5b + 7.5c = 3255
---------------------------subtracting eliminates b again:
1.5a + 0b + .5c = 457
1.5a + .5c = 457; another two unknown equation
:
Add our "two unknown" equations and find a:
a - .5c = 228
1.5a + .5c = 457
---------------------adding eliminates c
2.5a + 0c = 685
a = 685/2.5
a = 274 adults
:
Use a - .5c = 228 to find c, substitute 274 for a
274 - .5c = 228
-.5c = 228 - 274
-.5c = -46
c =-46/-.5
c = +92 old folks
:
I'm sure you can find the number of kids, equation 1 will work well:
 Check the solutions in eq 2 or 3.