Lesson Cramer's Rule
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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: {{{( matrix( 3,3,1,2,3,4,-5,6,7,8,-9 ))*( matrix( 3,1,x,y,z )) = ( matrix( 3,1,14,32,50 ))}}} Find the determinant, D, of the matrix {{{( matrix( 3,3,1,2,3,4,-5,6,7,8,-9 )) }}}: {{{D=det (matrix( 3,3,1,2,3,4,-5,6,7,8,-9 )) }}}. Then put the column {{{(matrix(3,1,14,32,50))}}} in place of the 1st, 2nd and 3rd columns sequentially, and calculate determinants as follows: {{{Dx = det (matrix(3,3,14,2,3,32,-5,6,50,8,-9))}}}, {{{Dy = det (matrix(3,3,1,14,3,4,32,6,7,50,-9))}}}, {{{Dz = det (matrix(3,3,1,2,14,4,-5,32,7,8,50))}}}. <B>Cramer's Rule</B> says: 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 <B>important note:</B> surely, this method assumes that determinant D is not equal to zero.
