How do we solve the 3 equations:
x+2y+3z = 14
4x-5y+6z = 32
7x+8y-9z = 50
Well, we can pick one of the variables in one of the equations and then substitute that into the other 2 equations and then solve those two equations simultaneously. A rather long winded algebraic solution.
A much better way, is to use the theory of determinants in matrices to help. This is Cramer's Rule.
First, rewrite the system of equations in a matrix form:
Find the determinant, D, of the matrix
Then put the column
in place of the 1st, 2nd and 3rd columns sequentially, and calculate determinants as follows:
x = Dx/D
y = Dy/D
z = Dz/D
Working these out, gives:
D = 354
Dx = 2652
Dy = 432
Dz = 480
x = 2652/354 = 7.4915257
y = 432/354 = 1.2203389
z = 480/354 = 1.3559322
An important note:
surely, this method assumes that determinant D is not equal to zero.
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