# Is there a unique solution for the equations x + y = 9, x + y + z = 10, x - y + z = 1

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### 1 Answer

You should eliminate the variable y from the first and the last equations such that:

`x + y + x - y + z = 9 + 1`

Reducing like terms yields:

`2x + z = 10`

You should eliminate the variable y from the second and the last equations such that:

`x + y + z + x - y + z = 10 + 1`

Reducing like terms yields:

`2x + 2z = 11`

You should solve now the system of two variables equations such that:

`{(2x + z = 10),(2x + 2z = 11):}`

You need to substract `2x + z = 10` from `2x + 2z = 11` such that:

`2x + 2z - 2x - z= 11 - 10`

`z = 1`

Substituting 1 for z in equation `2x + z = 10` yields:

`2x + 1 = 10 => 2x = 10 - 1 => 2x = 9 => x = 9/2`

Substituting 9/2 for x in the first equation `x + y = 9` yields:

`9/2+ y = 9 => y = 9 - 9/2 => y = 9/2`

**Hence, evaluating the solutions to the given system yields the unique solution `x = 9/2 , y = 9/2 , z = 1.` **