SOLUTION: I need help with an algebra word problem with system of equations. on the first day of school, 60% of the class were boys. During the school year six girls moved away and 6 boys

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Question 199118: I need help with an algebra word problem with system of equations.
on the first day of school, 60% of the class were boys. During the school year six girls moved away and 6 boys replaced them. At the end of the year, 75% of the class were boys. What is the number of boys and girls on the first day?
I've tried to do this:
60% of (x + y) = x
40% of (x + y) = y

Answer by jim_thompson5910(28550) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=# of boys and y=# of girls


Since there's only a choice between boys or girls, this means that there is a total of x%2By students. Also, because "on the first day of school, 60% of the class were boys", this means that 0.6%28x%2By%29=x (ie take 60% of the total and you get "x")


Also, since "During the school year six girls moved away and 6 boys replaced them." and "At the end of the year, 75% of the class were boys", we know that 0.75%28%28x%2B6%29%2B%28y-6%29%29=x%2B6 where "x+6" is the new count for the boys and "y-6" is the new count for the girls.


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0.6%28x%2By%29=x Start with the first equation.


0.6x%2B0.6y=x Distribute


10%280.6x%29%2B10%280.6y%29=10x Multiply EVERY term by 10 to make every number a whole number.


6x%2B6y=10x Multiply


6x%2B6y-10x=0 Subtract 10x from both sides.


-4x%2B6y=0 Combine like terms. So this is the new equation #1.


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0.75%28%28x%2B6%29%2B%28y-6%29%29=x%2B6 Start with the second equation.


0.75%28x%2B6%2By-6%29=x%2B6 Remove the inner parenthesis.


0.75%28x%2By%29=x%2B6 Combine like terms.


0.75x%2B0.75y=x%2B6 Distribute


100%280.75x%29%2B100%280.75y%29=100%28x%29%2B100%286%29 Multiply EVERY term by 100 to make every number a whole number.


75x%2B75y=100x%2B600 Multiply.


75x%2B75y-100x=600 Subtract 100x from both sides.


-25x%2B75y=600 Combine like terms. So this the new equation #2.







So we have the system of equations:

system%28-4x%2B6y=0%2C-25x%2B75y=600%29


Let's solve this system by use of substitution.


In order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

-4x%2B6y=0 Start with the first equation


6y=4x Add 4x to both sides


y=%284x%29%2F%286%29 Divide both sides by 6


y=%282%2F3%29x Reduce



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Since y=%282%2F3%29x, we can now replace each y in the second equation with %282%2F3%29x%2B0 to solve for x



-25x%2B75highlight%28%28%282%2F3%29x%29%29=600 Plug in y=%282%2F3%29x%2B0 into the second equation. In other words, replace each y with %282%2F3%29x%2B0. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.


-25x%2B%28150%2F3%29x=600 Multiply.


3%28-25x%29%2Bcross%283%29%28%28150%2Fcross%283%29%29x%29=3%28600%29 Multiply EVERY term by the LCD 3. This will eliminate the fractions.


-75x%2B150x=1800 Multiply.



75x=1800 Combine like terms on the left side


x=%281800%29%2F%2875%29 Divide both sides by 75 to isolate x


x=24 Divide


-----------------First Answer------------------------------


So the first part of our answer is: x=24.

This means that there are 24 boys (at the beginning of the school year)





Since we know that x=24 we can plug it into the equation y=%282%2F3%29x (remember we previously solved for y in the first equation).



y=%282%2F3%29x Start with the equation where y was previously isolated.


y=%282%2F3%29%2824%29 Plug in x=24


y=48%2F3 Multiply


y=16 Reduce.



-----------------Second Answer------------------------------


So the second part of our answer is: y=16 This means that there are 16 girls (at the beginning of the school year)





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Summary:


So our answers are: x=24 and y=16


This means that at the beginning of the school year, there are 24 boys and 16 girls. At the end of the school year, there are 30 boys and 10 girls.