X + Y = 10, X + Z = 20, Y + Z = 24, what is X + Y + Z = ?

Question

X + Y = 10

X + Z = 20

Y + Z = 24

what is X + Y + Z = ?

 

Please provide the steps in deriving your answer. Thanks.

Answer 1

Some of the other answers are good examples of solving a system of three equations in three unknowns, which is what this problem is asking. Though the simplest way to solve this problem is actually to notice that, if we sum the three equations, we get:

 

   X + Y = 10
   X + Z = 20
+  Y + Z = 24

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

2X + 2Y + 2Z = 54

 

Factoring out the 2, we have 2(X + Y + Z) = 54, and dividing both sides by 2 reveals that X + Y + Z = 27.

 

This won't always work when you're solving three equations in three unknowns, but it's a trick that shows up frequently enough on assignments, standardized tests, etc. that it's worth mentioning.

Answer 2

1)  X + Y = 10
2)  X + Z = 20
3)  Y + Z = 24
what is X + Y + Z = ?

 

From equation 1, x = 10-y

Plug 10-y in place of x in equation 2

 

10-y+z = 20   subtract 10 from both sides

-y + z = 10

 

We now have 2 equations with just y and z

 

-y + z = 10    new equation

 y + z = 24     add the 2 equations

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

     2z = 34

       z = 34/2

       z = 17

 

We now have the value of z.

Plug that into equation 3

 

y + z = 24

y + 17 = 24

y = 24-17

y = 7

 

We now have the value of both z and y.

Plug the value for y into equation 1 to find x

 

x + y = 10

x + 7 = 10

x = 10-7

x = 3

 

Your values are x = 3, y = 7, z = 17

 

Plugging those values into your original equations

you will see that they balance.

 

 

 

 

 

Answer 3

Hi Ezam.

 

The simplest way to do this is to set up a matrix. Here are the equations you'll set up:

 

X + Y + 0Z = 10

X + 0Y + Z = 20

0X + Y + Z = 24

 

So your matrix will look like:

 

1 1 0 10

1 0 1 20

0 1 1 24

 

Punch this into some matrix resolver like on Wolfram Alpha. The function you want to use is "rref" for "reduced row echelon form". Punch the following code into Wolfram:

 

rref([1,1,0,10],[1,0,1,20],[0,1,1,24])

 

Which will give you:

 

1 0 0 3

0 1 0 7

0 0 1 17

 

So X = 3, Y = 7, Z = 17.

 

X + Y + Z = 27

 

Hope this helps.

Answer 4

Given

X+Y = 10

X+Z = 20

Y+Z = 24

 

Let's restate the first two equations

Y = 10-X

Z = 20-X

 

And substitute these expressions of Y and Z in the 3rd equation

(10-X) + (20-X) = 24

 

Now rearrange terms and solve for X

-2X = 24-10-20 = -6

X = 3

 

Use this value of X to find Y and Z

Y = 10-X = 10-3 = 7

Z = 20-X = 20-3 = 17

 

Since X = 3, Y = 7 and Z = 17

X + Y + Z = 3+7+17 = 27