SOLUTION: The sum of three integers is 54. The first integer is twice the difference between the second and third integers. The second is 38 more than twice the third. Find the integers.

Question 468170: The sum of three integers is 54. The first integer is twice the difference between the second and third integers. The second is 38 more than twice the third. Find the integers.
Answer by karaoz(32) (Show Source): You can put this solution on YOUR website!

Hi there!
I can see you have three integers and I do not know what they are but you are telling me to find them. I will!

First thing I will do is to name these integers using a single letter - a different letter for different integer. Using one and the same letter for all integers would not work since this would automatically mean that the integers have to be the same. However, using different letters will work even in the case when some of your numbers happen to be really the same. In that case all I will end up with is that two or more of my letters will end up being assigned the same value... So, names then.

Let the first number be named a!
Let the second number be named b!
Let the third number be named c!

Now I will rephrase the clues you have given me by using these names instead of "first number", "second number" or "third number". Watch me!

Original problem statements / Translation

"The sum of three integers is 54. "
" a + b + c = 54"
"The first integer is twice the difference between the second and third integers."
"a = 2(b - c)"
"The second is 38 more than twice the third."
"b = 2c + 38"
"Find the integers."
"Find a, b and c."

The rest is the matter of solving the system of 3 equations with 3 unknowns.
The system is:

a + b + c = 54
        a = 2(b - c)
        b = 2c + 38

We can see that b is already expressed through c only (3rd equation) and if we use that to substitute in the second equation we will also have a expressed only in terms of c. Taking these two together and replacing them in the first equation, we will get only one equation with c being the only unknown. So, it goes then:

b = 2c + 38
a = 2(2c + 38 - c) = 2(c + 38) = 2c + 76

The equation to solve is

2c + 76 + 2c + 38 + c =  54 
             114 + 5c =  54
                   5c = -60
                    c = -12

Using this result to get a and b:

b = 2c + 38
b = 2(-12) + 38 = 14

and

a = 2c + 76
a = 2(-12) + 76
a = -24 + 76
a = 52

Done! a = 52, b = 14, c = -12, or
First number is 52
Second number is 14
Third number is -12.

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