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Simultaneous Equations •Elimination Method •Substitution method •Graphical Method •Matrix Method What are they? • Simply 2 equations – With 2 unknowns – Usually x and y • To SOLVE the equations means we find values of x and y that – Satisfy BOTH equations [work in] – At same time [simultaneously] Elimination Method A B We have the same number of y’s in each 2x – y = 1 If we ADD the equations, the y’s disappear + 3x + y = 9 5x = 10 =2 x 2x2–y=1 4–y=1 y=3 Divide both sides by 5 Substitute x = 2 in equation A Answer x = 2, y = 3 Elimination Method A B 5x + y = 17 3x + y = 11 2x =6 =3 x 5 x 3 + y = 17 15 + y = 17 y=2 We have the same number of y’s in each If we SUBTRACT the equations, the y’s disappear Divide both sides by 2 Substitute x = 3 in equation A Answer x = 3, y = 2 Elimination Method A B 2x + 3y = 9 2x + y = 7 2y = 2 y =1 2x + 3 = 9 2x = 6 x=3 We have the same number of x’s in each If we SUBTRACT the equations, the x’s disappear Divide both sides by 2 Substitute y = 1 in equation A Answer x = 3, y = 1 Elimination Method A B 4x - 3y = 14 + 2x + 3y = 16 6x = 30 x =5 20 – 3y = 14 3y = 6 y=2 We have the same number of y’s in each If we ADD the equations, the y’s disappear Divide both sides by 6 Substitute x = 5 in equation A Answer x = 5, y = 2 Basic steps • Look at equations • Same number of x’s or y’s? • If the sign is different, ADD the equations otherwise subtract tem • Then have ONE equation • Solve this • Substitute answer to get the other • CHECK by substitution of BOTH answers What if NOT same number of x’s or y’s? A B A 3x + y = 10 5x + 2y = 17 If we multiply A by 2 we get 2y in each - 6x + 2y = 20 B 5x + 2y = 17 x =3 In B 5 x 3 + 2y = 17 15 + 2y = 17 y=1 Answer x = 3, y = 1 What if NOT same number of x’s or y’s? A B 4x - 2y = 8 3x + 6y = 21 If we multiply A by 3 we get 6y in each A 12x - 6y = 24 B 3x + 6y = 21 15x = 45 x=3 In B 3 x 3 + 6y = 21 6y = 12 y=2 + Answer x = 3, y = 2 …if multiplying 1 equation doesn’t help? A 3x + 7y = 26 B 5x + 2y = 24 A 15x + 35y = 130 B 15x + 6y = 72 29y = 58 y=2 In B 5x + 2 x 2 = 24 5x = 20 x=4 Multiply A by 5 & B by 3, we get 15x in each - Could multiply A by 2 & B by 7 to get 14y in each Answer x = 4, y = 2 …if multiplying 1 equation doesn’t help? A 3x - 2y = 7 B 5x + 3y = 37 A 9x – 6y = 21 B 10x + 6y = 74 = 95 19x x=5 In B 5 x 5 + 3y = 37 3y = 12 y=4 Multiply A by 3 & B by 2, we get +6y & -6y + Could multiply A by 5 & B by 3 to get 15x in each Answer x = 5, y = 4 Substitution Method Given the following equations : y = x + 3 (i) y = 2x (ii) Replace the y in equation (i) with 2x from equation (ii) 2x = x + 3 2x – x = 3 x=3 Sub. x = 3 into either of the two original equations to find the value of y y = x + 3 (i) y=3+3 y=6 The answer is (3, 6) Substitution Method A tool hire firm offers two ways in which a tool may be hired: •Plan A - $20 a day •Plan B - A payment of $40 then $10 a day Find the number of days whereby there is no difference in the cost of hiring the tool from Plan A and Plan B. Let : y = $ in hiring tool x = no. of days hiring tool y = 20x -----(1) y = 40 + 10x -----(2) Sub.(1) into (2) 20x = 40 + 10x 10x = 40 x = 4, y = 80 Graphical Method x+y=6 Let x = 0 y=6 Coordinates (0, 6) Let y = 0 x=6 Coordinates (6, 0) 2x + y = 8 Let x = 0 y=8 Coordinates (0, 8) Let y = 0 2x = 8 x=4 Coordinates (4, 0) 1. x + y = 6 2x + y = 8 Graphical Method y=x+3 Let x = 0 y=3 Coordinates (0, 3) Let y = 0 x = -3 Coordinates (-3, 0) y = 2x Let x = 0 y=0 Coordinates (0, 0) Let y = 4 2x = 4 x=2 Coordinates (2, 0) y=x+3 y = 2x