Simultaneous Equations OCR Module 8 What are they

Simultaneous Equations OCR Module 8

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 2 x – y = 1 If we ADD the equations, the y’s disappear + 3 x + y = 9 5 x = 10 =2 x 2 x 2–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 5 x + y = 17 3 x + y = 11 2 x =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 2 x + 3 y = 9 2 x + y = 7 2 y = 2 y =1 2 x + 3 = 9 2 x = 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 4 x - 3 y = 14 + 2 x + 3 y = 16 6 x = 30 x =5 20 – 3 y = 14 3 y = 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 3 x + y = 10 5 x + 2 y = 17 If we multiply A by 2 we get 2 y in each - 6 x + 2 y = 20 B 5 x + 2 y = 17 x =3 In B 5 x 3 + 2 y = 17 15 + 2 y = 17 y=1 Answer x = 3, y = 1

What if NOT same number of x’s or y’s? A B 4 x - 2 y = 8 3 x + 6 y = 21 If we multiply A by 3 we get 6 y in each A 12 x - 6 y = 24 B 3 x + 6 y = 21 15 x = 45 x=3 In B 3 x 3 + 6 y = 21 6 y = 12 y=2 + Answer x = 3, y = 2

…if multiplying 1 equation doesn’t help? A 3 x + 7 y = 26 B 5 x + 2 y = 24 A 15 x + 35 y = 130 B 15 x + 6 y = 72 29 y = 58 y=2 In B 5 x + 2 x 2 = 24 5 x = 20 x=4 Multiply A by 5 & B by 3, we get 15 x in each - Could multiply A by 2 & B by 7 to get 14 y in each Answer x = 4, y = 2

…if multiplying 1 equation doesn’t help? A 3 x - 2 y = 7 B 5 x + 3 y = 37 A 9 x – 6 y = 21 B 10 x + 6 y = 74 = 95 19 x x=5 In B 5 x 5 + 3 y = 37 3 y = 12 y=4 Multiply A by 3 & B by 2, we get +6 y & -6 y + Could multiply A by 5 & B by 3 to get 15 x in each Answer x = 5, y = 4