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Simultaneous Linear Equations – A General Solution Mathematical Mischief – January 2013 Question - Take the following simultaneous linear equations: ax + by = c dx + ey = f Devise a general solution for x and y . ax + by = c ax + by = ax = x = dx + ey = f dx + ey = c c − by dx c − by x a c − by f − ey = a d f = f − ey = f − ey d c − by = a d(c − by) = f − ey d a( f − ey) cd − bdy = af − aey cd − af = bdy − aey cd − af = (bd − ae)y cd − af bd − ae = Step 1: State both equations separately. Step 2: Solve each equation for x . Step 3: Substitute x for the solution of the other equation. Step 4: Solve for y . y As it would be inappropriate to substitute the value of y into one equation to solve for x , it is far more reasonable to start again, solving the simultaneous equations from the beginning. ax + by = by = y = c c − ax dx + ey = ey c − ax y b c − ax f − dx = b e f = f − dx = f − dx e Step 5: Solve each equation for y . Step 6: Substitute y for the solution of the other equation. January 2013 – http://www.mathematicalmischief.com/ - @mathmischief c − ax = b e(c − ax) = f − dx e b( f − dx) ce − aex = bf − bdx ce − bf = aex − bdx ce − bf = (ae − bd)x ce − bf ae − bd = Step 7: Solve for x . x As a result, when solving two simultaneous equations of the form: ax + by = c dx + ey = f We obtain a general solution that: x= ce − bf ae − bd and y= cd − af bd − ae January 2013 – http://www.mathematicalmischief.com/ - @mathmischief