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For matinees,a movie theater charges $7 per adult and $4 per child.if 84 people attended a matinee and paid a total of $432,How many adults and how many children were there .
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7 Answers

 First choose to let x=number of adults and y = number of children.
Then you can write 2 equations from the information given.
1)  7x + 4y = 432
2)  X + y = 84
Solve equation 2 for x:   X = 84 – y
Substitute this value into equation 1:

7(84-y) + 4y = 432
588-7y + 4y =432
-3y = 432-588 = - 156
Y = -156/-3 = 52
So x= 84-52 = 32
You know that each adult if charged $7 and each child is charged $4 for a matinee show.
You also know that 84 people attended and that a total of $432 was paid, so...
If a = # of adults and c = # of children,
a + c = 84
$7*a +$4*c = 432
Substitute one of the variables and solve.
a + c = 84
     -c     -c
      a = 84 - c
Plug this equation for "a" in the second equation.
7*(84 - c) + 4c = 432
588 - 7c + 4c = 432
 588 - 3c = 432
-588         -588
       -3c  = -156
       -3          -3
         c =  52
So 32 children attended the matinee.
a + 52 = 84
     -52   -52
        a = 32
And 32 adults attended the matinee.
C + A = 84
7A + 4C = 432
Rewriting the first equation C=84-A and substituting in the 2nd equation. 
7A + 4 (84-A) = 432. Now distributing
7A + 336 -4A=432. Combining like terms after rearranging. 
3A=96. Divide both sides by 3. 
A= 32. Substituting A in 1st equation yields
so C=52.
Checking the 2nd equation to verify
7(32)+4(52)=224+208=432. Checks
There are two unknowns (# adults and # children) so you need two equations to solve it. If you look carefully you will see two pieces of information:
The total # of people (84)
The total amount of money collected $432
If you let A be the # of adults and C be the # of children then:
A*$7+C*$4=$432 (remember the total amount of $ was $432)
Also A+C=84 (there were 84 total people)
So Given:
A*7+C*4=432    (eqn 1)
A+C=84              (eqn 2)
We can substitute
A=84-C  (from eqn 2)
-3C= -156
and since A+C=84
A=84-52 = 32
Let's break the word problem down.
Let a=number of adults and c=number of children
We know two things:
1. The total number of people in attendance equals 84 or a+c=84
2. The total take is $432, comprised of adults each paying $7 and children each paying $4.  This can be expressed algebraically as 7a+4c=432
We know have two equations, with two unknowns (a & c), which can be solved, as follows:
From the first equation, we can easily isolate a on the left side:  a=84-c
We can now substitute (84-c) for a in the second equation, and solve for c, as follows:
   c=52 (this is the number of children)
Going back to the first equation, since the total number of people attending is 84,
a=32 (this is the number of adults)
You can now check your work, by substituting for a and c in the second equation, to see if both sides come out equal:
    432=432 (check!)
Let me know if you have any questions.
George T.
Hi Ma,
This is a simple system of equations.  The tricky part is translating the word problem into the equations in the first place.
So we know two different things about this matinee showing - we know the total ticket sales, and we know how many people attended.  Those are going to be your two different equations.  We also know that we're looking for two different quantities; the number of adults present and the number of children present.  Since we don't know those yet, we'll call them x and y.  I'm calling the number of adults x and the number of children y.
Let's look at the second thing first - we know that 84 people attended the matinee.  That means that the number of adults plus the number of children equals 84.  So:
x + y = 84
That's your first equation.  Now for the other one, the ticket sales, we know that the theater made 7 on every adult and 4 on every child, and that the total adds up to 432.  So:
7x + 4y = 432
Once you've got those two equations, now we have to solve the system.  To do that, just remember that you have to keep switching equations each time through.  First, let's solve our first equation for y:
y = 84 - x
Now that we know a value for y in terms of x, plug that quantity in for y in the other equation.  (This is the switching part.) So:
7x + 4y = 432
7x + 4(84 - x) = 432
Now just solve for x.  Bear with me; here we go:
7x + 336 - 4x = 432
3x + 336 = 432
3x = 96
x = 32
So there were 32 adults.  Now it's time to switch equations once more, to find the number of children.  Plug that new x value into the first equation:
32 + y = 84
y = 52
So there were 52 children.  Must have been a Disney movie!
Hope this helps!
Here, you have two unknowns - the number of adults (call that X) and the number of children (call that Y), and two equations, so solve the equations for X and Y.
X + Y = 84
7X + 4Y = 432
Eliminate one of the variables, say, Y by multiplying the first equation by 4 and subtracting it from the second:
  7X + 4Y = 432
-(4X + 4Y = 84*4 = 336)
  3X = 432 - 336 = 96  Divide by 3 to get the value of X
    X = 32
Now substitute that value of X in the first equation:
32 + Y = 84
Y = 84 - 32 = 52
So 32 adults and 52 children were there.