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.

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) (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 and

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

Start with the given system of equations:

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

Start with the first equation

Subtract from both sides

Rearrange the equation

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

Since , we can now replace each in the second equation with to solve for

Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.

Distribute to

Multiply

Combine like terms on the left side

Add 198 to both sides

Combine like terms on the right side

Divide both sides by 12 to isolate x

Divide

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

So the first part of our answer is:

Since we know that we can plug it into the equation (remember we previously solved for in the first equation).

Start with the equation where was previously isolated.

Plug in

Multiply

Combine like terms

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

So the second part of our answer is:

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

So our answers are:

and

This means that the smaller number is 19 and the larger number is 80
Answer by jonvaliente(64) (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.

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