# Given that the sum of three numbers is forty-nine. The sum of the first and third number is nine less than the second, and the third number is a quarter of the second. Derive a system of equations...

Given that the sum of three numbers is forty-nine. The sum of the first and third number is nine less than the second, and the third number is a quarter of the second.

Derive a system of equations representing the information given.

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### 2 Answers

Let's call those three numbers `x,y` and `z.` Their sum is 49 hence

`x+y+z=49`

Sum of the first and third is nine less than the second hence

`x+z=y-9`

The third is quarter of the second hence

`z=y/4`

So you have system of three equations with three variables.

`x+y+z=49`

`x-y+z=-9`

`y=4z`

Now you can solve this system by using Gauss elimination or some other method.

**Your final solution is **`x=51/4,` `y=29` **and** `z=29/4.`

`x+y+z=49` (1)

`x+z=y-9` (2)

`z=1/4 y` (3)

`x+y+z=49` (4)

`x-y+z=-9` (5)

`z=1/4y` (6)

Subtracting (5) from (4):

`2y=58` `rArr y=29`

`z=29/4`

so from (1):

`x+29+29/4=49`

`x+ 145/4 =49`

`x=49-145/4=51/4`

so the solutions are:**`x=51/4;y=29;z=29/4` **