Answers:
Number of adults = 79
Number of children = 29
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Work Shown:
Let
x = number of adults
y = number of children
There are 108 total people, so x+y = 108. We can subtract x from both sides to isolate y and we get y = 108-x. We will use this equation later.
If we charge $8 per adult, then 8x represents all of the money pulled in from just the adults attending (ignoring the children for now)
Each child ticket costs $5, so 5y is the total money brought in from all the child ticket sales (ignore the adults temporarily)
In total, 8x+5y is the grand total for everyone that attends (adults and children). This grand total is given at 777 dollars.
Therefore, we know the second equation is 8x+5y = 777
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Lets use substitution to eliminate the variable y
8x+5y = 777
8x+5( y ) = 777
8x+5( 108-x ) = 777 .... y replaced with 108-x
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From here let's solve for x
8x+5( 108-x ) = 777
8x+5(108) + 5(-x) = 777 .... distributive property
8x + 540 - 5x = 777
8x - 5x + 540 = 777
3x + 540 = 777
3x + 540-540 = 777-540 .... subtracting 540 from both sides
3x = 237
3x/3 = 237/3 .... dividing both sides by 3
x = 79
Use this to find y
y = 108-x
y = 108-79 ... x is replaced with 79
y = 29
The two answers are x = 79 and y = 29. Meaning that 79 adults showed up along with 29 children.
Note how x+y = 79+29 = 108 which partially confirms our two answers. Plug (x,y) = (79,29) into the expression 8x+5y and you should find that the result is 777, which fully confirms the ordered pair solution.