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Gaussian Elimination to solve systems of linear equations Consider 2 simultaneous equations with 2 unknowns: 5x +3y = 11 ………..(1) 2x - 2y = -2……….. (2) Divide equation (1) by 5, ie x + 3y = 11 ………..(3) 5 5 multiply equation (3) by 2 and then subtract it from equation (2) 2x - 2y = -2 - 2x + 6y = 22 5 5 0 - 16y = -32 , ie y =2 , and hence x = 1 5 5 The procedure above to solve the equations forms the basis of a general procedure using a matrix ( a rectangular array of numbers). In this case the matrix is an augmented matrix, whereby the equation coefficients ( ann) and constants( kn) are shown ie [ a11 a12 k1 ] or [ 5 3 11] [ a21 a22 k2 ] [ 2 -2 -2 ] Divide the first row by 5, ie [ 1 3/5 11/5 ] Now subtract 2x the first row (ie [ 2 6/5 22/5 ] from the second row to give a zero in the first column of the second row ie [ 1 3/5 11/5 ] [ 0 -16/5 -32/5] Now divide the second row by -16/5 ie [ 0 1 2 ] ie y = 2 To obtain x multiply the second row by -3/16 and subtract from row 1 Ie [ 1 3/5 11/5 ] = [ 1 0 1] ie x = 1 - [ 0 3/5 6/5 ] The use of a matrix to solve 2 simultaneous equations with 2 unknowns is cumbersome but forms the basis of a general procedure, called Gaussian Elimination to solve n simultaneous equations with n unknowns Examples Use matrices to solve the following equations by elimination (a) x + y = 3, 2x - y = 3 (b) 4x + y = 7, 5x – y = 2 (c) x + 2y = 1, -x + 2y = 3, (d) 2x - 3y =7, x + y = 1 . Use Gaussian Elimination to find the values of x, y and z in the simultaneous equations: (e) 2x + 3y - 2z = 6, 3x + 2y + 3z = 3, 4x - 5y + 7z = 6 (f) x + 2y + 2z = 7, 2x + y + z = 5, 3x - 2y + 2z = 5