# What is the solution of the equations x + y + z = 18, x + y - z = 8 and x - y + z = 10

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### 2 Answers

This is a system of three linear equations with three unknowns. There are several ways of solving the system, but the easiest one is to solve by parts two by two, that is, first take two of them and solve for one unknown and then proceed from that. Just looking at the equations, the easiest way to start solving them is to take the second and third equations and sum them both sides:

`x+y-z=8`

`x-y+z=10`

`2x = 18`

`x=9` .

If now we add the first and the second equations we get

`2x+2y=26`

`or x+y=13`

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Since x is already calculated above, it follows that

`9+y=13 or y=13-9=4`

Now, we may use any one of the equations to calculate z. From 3rd equation, for example, we have:

`x-y+z=10 or 9-4+z=10 or z=10-9+4=5`

Finally,. collecting all answers:

(x,y,z) = (9,4,5).

We can check that all three equations are satisfied by this.

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The set of equations x + y + z = 18, x + y - z = 8 and x - y + z = 10 have to be solved for x, y and z.

x + y + z = 18 ...(1)

x + y - z = 8 ...(2)

x - y + z = 10 ...(3)

Add (2) and (3)

x + y - z + x - y + z = 8 + 10

=> 2x = 18

=> x = 9

Add (1) and (2)

x + y + z + x + y - z = 26

Substitute x = 9

=> 18 + 2y = 26

=> y = 4

Substitute x = 9 and y = 4 in (1)

z = 18 - 9 - 4 = 5

**The solution of the given system of equations is x = 9, y = 4 and z = 5**