SOLUTION: The sum of two numbers is 99. If 10 times the smaller is decreased by 2 times the larger, the result is 30. Find the numbers.

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Question 189716: The sum of two numbers is 99. If 10 times the smaller is decreased by 2 times the larger, the result is 30. Find the numbers.
Found 2 solutions by jim_thompson5910, jonvaliente:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let
x = smaller number
y = larger number



"The sum of two numbers is 99" translates to x%2By=99 and

"If 10 times the smaller is decreased by 2 times the larger, the result is 30" translates to 10x-2y=30



Start with the given system of equations:

system%28x%2By=99%2C10x-2y=30%29


Let's solve by substitution


Now 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

x%2By=99 Start with the first equation


y=99-x Subtract x from both sides


y=-x%2B99 Rearrange the equation



---------------------

Since y=-x%2B99, we can now replace each y in the second equation with -x%2B99 to solve for x



10x-2highlight%28%28-x%2B99%29%29=30 Plug in y=-x%2B99 into the second equation. In other words, replace each y with -x%2B99. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



10x%2B%28-2%29%28-1%29x%2B%28-2%29%2899%29=30 Distribute -2 to -x%2B99


10x%2B2x-198=30 Multiply


12x-198=30 Combine like terms on the left side


12x=30%2B198Add 198 to both sides


12x=228 Combine like terms on the right side


x=%28228%29%2F%2812%29 Divide both sides by 12 to isolate x



x=19 Divide





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


So the first part of our answer is: x=19









Since we know that x=19 we can plug it into the equation y=-x%2B99 (remember we previously solved for y in the first equation).



y=-x%2B99 Start with the equation where y was previously isolated.


y=-%2819%29%2B99 Plug in x=19


y=-19%2B99 Multiply


y=80 Combine like terms



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


So the second part of our answer is: y=80









-----------------Summary------------------------------

So our answers are:

x=19 and y=80


This means that the smaller number is 19 and the larger number is 80

Answer by jonvaliente(64) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = larger number
y = smaller number
The sum of two numbers is 99:
x+y=99 (1)
10 times the smaller is decreased by 2 times the larger is 30
10y-2x=30 (2)
Multiplying (1) by 2, we get:
2x+2y=198 (1)
Adding (1) and (2), we get:
10y+2y-2x+2x=228
12y=228
y=19
x+19=99
x=80
The two numbers are 19 and 80.