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Solution - 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. - HSC Science (Computer Science) 12th Board Exam - Mathematics and Statistics

Question

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 sum of 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.

Solution

Given that the sum of three numbers, x, y and z is 6.
From the given statement, we have,
3 (x+z)- y=10
5 (x+y)- 4z= 3
Thus, the system of equations are:
x+y+z=6
3x-y+3z=10
5x+5y-4z=3

Let us write the above equations in the matrix form as:

`[[1,1,1],[3,-1,3],[5,5,-4]][[x],[y],[z]]=[[6],[10],[3]]`

AX=B

`A=[[1,1,1],[3,-1,3],[5,5,-4]],X=[[x],[y],[z]],B=[[6],[10],[3]]`

Thus ,`[[1,1,1],[3,-1,3],[5,5,-4]]A^-1=[[1,0,0],[0,1,0],[0,0,1]]`

Thus ,`[[1,1,1],[3,-1,3],[5,5,-4]]A^-1=[[1,0,0],[0,1,0],[0,0,1]]`

Thus ,`[[1,1,1],[3,-1,3],[5,5,-4]]A^-1=[[1,0,0],[0,1,0],[0,0,1]]`

where `A=[[1,1,1],[3,-1,3],[5,5,-4]],X=[[x],[y],[z]] and B=[[6],[10],[3]]`

Thus, A-1 exists
We know that AA-1 = I

`"Thus " ,[[1,1,1],[3,-1,3],[5,5,-4]]A^-1=[[1,0,0],[0,1,0],[0,0,1]]`

Applying R2→ R2 - 3R1 , we have 

`[[1,1,1],[0,-4,0],[5,5,-4]] A^-1=[[1,0,0],[-3,1,0],[0,0,1]]`

 Applying R3→ R3- 5R1 , we have

`[[1,1,1],[0,-4,0],[0,0,-9]] A^-1=[[1,0,0],[-3,1,0],[-5,0,1]]`

 Applying R3 →R3/-9, we have

`[[1,1,1],[0,-4,0],[0,0,1]] A^-1=[[1,0,0],[-3,1,0],[-5/9,0,-1/9]]`

Applying R2→R2/4 , we have

`[[1,1,1],[0,1,0],[0,0,1]] A^-1=[[1/4,1,0],[3/4,-1/4,0],[5/9,0,-1/9]]`

 Applying R1→ R1- R2 , we have

`[[1,0,1],[0,1,0],[0,0,1]] A^-1=[[1/4,1/4,0],[3/4,-1/4,0],[5/9,0,-1/9]]`

Applying R1→R1-R3 , we have

`[[1,0,0],[0,1,0],[0,0,1]] A^-1=[[11/36,1/4,1/9],[3/4,-1/4,0],[5/9,0,-1/9]]`

AX=B

A-1 AX=A-1 B
IX=A-1 B

X=A-1 B

`X=[[11/36,1/4,1/9],[3/4,-1/4,0],[5/9,0,-1/9]][[6],[10],[3]]`

`[[x],[y],[z]]=[[1],[2],[3]]`

Thus, the numbers are 1, 2 and 3.

 

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 2014-2015 (March) (with solutions)
Question 2.2.3 | 4 marks

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Reference Material

Solution for question: 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. concept: null - Elementary Operation (Transformation) of a Matrix. For the courses HSC Science (Computer Science), HSC Science (General) , HSC Arts, HSC Science (Electronics)
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