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matrix method

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2 Answers

Write the system first in matrix form:
 
((1,2,-3),(2,4,-5),(3,-1,1))*(x,y,z)=(4,12,3)
 
Next, check the determinant of the matrix. If it is zero, the system has no solution (or, if the right side was zero, infinitely many solutions).
 
In this case, the determinant is non-zero:
 
det ((1,2,-3),(2,4,-5),(3,-1,1))=7,
 
which means it the system has a unique solution. You find this solution by computing the inverse of the matrix,
 
((1,2,-3),(2,4,-5),(3,-1,1))-1=1/7*((-1,1,2),(-17,10,-1),(-14,7,0))
 
and multiplying it by (4,12,3). You get
 
(x,y,z) = (2,7,4).
If you have a system of equations, you can write it in the form:
 
Ax=b; here A and b are the matrix of the coefficients and is the column vector on the right side. The solution is given by:
 
x=A-1b where
 
A-1 is the matrix inverse of the original matrix A. If A-1 does not exist, then there is no solution, the system is inconsistent.
 
So, find the matrix inverse to your matrix, then multiply it by (4, 12, 3) vector and the resulting vector will give you the solution in the form (x,y,z).
 
I suspect you made a typo in the first equation, so I can't proceed further to illustrate this. Hope you can do it yourself and you know how to invert a matrix.