SOLUTION: I am having trouble with understanding this question: The sum of three numbers is 40. The second number is twice the first number. The sum of the first two numbers is 2 more tha

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Question 1171400: I am having trouble with understanding this question:
The sum of three numbers is 40. The second number is twice the first number. The sum of the first two numbers is 2 more than the third number. What are the three numbers?
So far, I've worked out:
x+y+z=10
y=2x
x+y=z+2
Found 5 solutions by Alan3354, MathLover1, ikleyn, MathTherapy, greenestamps:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of three numbers is 40. The second number is twice the first numxber. The sum of the first two numbers is 2 more than the third number. What are the three numbers?
So far, I've worked out:
x+y+z=40
y=2x
x+y=z+2
====================
Your eqns are correct except for the sum = 10
Sub for y in the 1st and 3rd eqns.
---
x + 2x + z = 40 ---> 3x + z = 40
x + 2x = z + 2 ---> 3x = z + 2 --> 3x - z = 2
------------
3x + z = 40
3x - z = 2
------------------- Add
6x = 42
x = 7
--------
Solve for y, then solve for z

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

The sum of three numbers is 40.
x%2By%2Bz=40......eq.1
The second number is twice the first number.
y=2x......eq.2
The sum of the first two numbers is 2 more than the third number.
x%2By=z%2B2......eq.3

x%2By%2Bz=40......eq.1, substitute y from eq.2
x%2B2x%2Bz=40
3x%2Bz=40....solve for z
z=40-3x............eq.1a
go to
x%2By=z%2B2......eq.3,substitute y from eq.2 and z from eq.1a
x%2B2x=40-3x%2B2
3x=42-3x
3x%2B3x=42
6x=42
x=7
go to eq.2
y=2x......eq.2
y=2%2A7
y=14
go to eq,1a
z=40-3x............eq.1a
z=40-3%2A7
z=40-21
z=19
the three numbers are:7, 14, and 19

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.

            This problem has a simple,  short,  straightforward and elegant solution.

            With my magic lamp,  I will show it to you now.

            Watch attentively every my step. 


Let x be the first number.


Then the second number is 2x, according to the condition.


Then the third number is  (40-x-2x) = 40-3x.



The sum of the first two numbers is 2 more than the third number

    x + 2x = (40-3x) + 2



Solve this simple equation

    x + 2x + 3x = 40+2

        6x      = 42

         x      = 42/6 = 7.



ANSWER.  The first number is 7;  the second number is 2*7 = 14, and the third number is  40 - 7 - 14 = 19.

Solved.

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

It is how this problem  SHOULD  be treated,  solved, explained and taught.

The problem is designed, intended and expected to be solved in this way.

Any other way is  LAME.

I know it and know this way,  because I have my magic lamp.

A miracle is the fact that the problem is solved using only one single unknown and one single equation.

It is intended to teach 6-th grade (7-th grade ?) students in this way.



Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

I am having trouble with understanding this question:
The sum of three numbers is 40. The second number is twice the first number. The sum of the first two numbers is 2 more than the third number. What are the three numbers?
So far, I've worked out:
x+y+z=10
y=2x
x+y=z+2
The setup of the equations is fine, with the exception of the 1st, which has a sum of 40, not 10..
x + y + z = 40 ------ eq (i)
y = 2x ------- eq (ii)
x + y = z + 2_____x + y - z = 2 ------- eq (iii)
2z = 38 ------ Subtracting eq (iii) from eq (i) 
highlight_green%28matrix%281%2C5%2C+z%2C+%22=%22%2C+38%2F2%2C+%22=%22%2C+19%29%29

2x + 2y = 42 ------ Adding eq (iii) & (i) 
  x + y = 21 ------ eq (iv)
 x + 2x = 21 ------ Substituting 2x for y in eq (iv)
     3x = 21
highlight_green%28matrix%281%2C5%2C+x%2C+%22=%22%2C+21%2F3%2C+%22=%22%2C+7%29%29

highlight_green%28matrix%281%2C5%2C+y%2C+%22=%22%2C+2%287%29%2C+%22=%22%2C+14%29%29 ----- Substituting 7 for y in eq (ii)

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


As tutor @ikleyn says, the easiest way to solve this problem is using a single variable.

The path she used to the solution is a good one; here is an alternative.

Learn from this that there are nearly always different ways to set up a problem for solving.

As you are learning mathematics, always explore different ways of solving problems and find ways that "work" for you.

Start the same way she does: the first number is x, so the second is 2x.

Now go a different direction than she did. "the sum of the first two numbers is 2 more than the third number" --> "the third number is 2 less than the sum of the first two": the third number is 3x-2.

Now the equation to be solved says that the sum of the three numbers is 40:

x%2B2x%2B%283x-2%29=40
6x-2+=+40
6x+=+42
x+=+7

ANSWERS: The three numbers are
x = 7
2x = 14
3x-2 = 19