The sum of three numbers is 9. If we multiply third number by 3 and add to the second number, we get 16. By adding the first and the third number and then subtracting twice the second number from this sum, we get 6. Use this information and find the system of linear equations. Hence, find the three numbers using matrices.

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#### Solution

Let the three numbers be x, y, z

From the first condition

x + y + z = 9

From the second condition

y + 3z = 16

From the third condition

x – 2y + 2 = 6

x + y + z = 9

y + 3z = 16

x – 2y + z = 6

`[[1,1,1],[0,1,3],[1,-2,1]][[x],[y],[z]]=[[9],[16],[6]]`

`R_3-R_1`

`[[1,1,1],[0,1,3],[0,-3,0]][[x],[y],[z]]=[[9],[16],[-3]]`

`x+ y + z = 9`

`y = 3z = 16 `

`-3y=-3 =>y=1`

`1+3z=16`

`z=5`

`x+1+5=9`

`x=3`

`therefore x=3, y=1, z=5`

Concept: Elementary Transformations

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