The sum of three numbers is 6 . When second number is subtracted from thrice the sum of first and third number , we get number 10. Four times the third number is subtracted from five times the sum of first and second number , the result is 3 . Using above information, find these three numbers by matrix method.

Explanation

Let the numbers be x, y and z
According to question, ` x + y + z = 6` …(i)
` 3 (x +z) - y = 10 `
or ` 3x - y + 3z = 10` …(ii)
Also ` 5(x + y) - 4z = 3`
or ` 5x + 5y - 4z = 3` ….(iii)
Thus `[(1,1,1),(3,-1,3),(5,5,-4)][(x),(y),(z)]=[(6),(10),(3)]`
Applying `R_(3) to R_(3) - 5 R_(1)`
`[(1,1,1),(3,-1,3),(0,0,-9)][(x),(y),(z)]=[(6),(10),(-27)]`
Applying `R_(2) to R_(2) - 3 R_(1)`
`[(1,1,1),(0,-4,0),(0,0,-9)][(x),(y),(z)]=[(6),(-8),(-27)]`
Thus ` x + y + z = 6` ...(i)
` -4y = - 8`
or ` y = 2` ..(iv)
and ` -9z = - 27`
or ` z = 3` ....(v)
Thus ` x = 6 - (2+3)`
` = 6 - 5 = 1`
Hence, x = 1 , y = 2 and z = 3