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Simultaneous equations Simultaneous means at the same time To solve two equations simultaneously is to find a point which is on both graphs. This is the point of intersection of the two graph lines y Example y = 2x + 4 Solve these simultaneous equations y=? y = 2x + 4 y = -0.5x + 10 or y = -0.5x + 10 Find the point of intersection of the lines y = 2x + 4 and y = -0.5x + 10 x x=? How? Substitution method: put one equation “into” the other equation Elimination method: add or subtract the two equations to eliminate one variable Elimination method Linear equations can be written in different ways: eg y = 2x – 3 y – 2x = -3 y – 2x + 3 = 0 If both equations are written in the same form the elimination method is best 2x – y = 8 x +y=4 Equations Eliminate y 2x – y = 8 Add x +y=4 3x = 12 x = 4 One solution x +y=4 4+y=4 y=0 Substitute Other solution 2x – y = 8 Check! 2x4 – 0 = 8? 8 = 8 Yes Steps: 1) Line up the terms: x, y, constant, and = 2) Select one variable to eliminate whatever variable has the same number in each equation – scaling may be needed! 3) Do we add or subtract the equations? Add if signs different, Subtract if same sign 4) Solve to find one variable 5) Substitute this variable back into one equation to find the other variable Solution: x = 4, y = 0 or as a coordinate (4,0) 6) Substitute both solutions into the other original equation to check answer is correct Elimination method A Eg 1) 3x + y = 13 2x – y = 7 Theta p39 Ex 5.2 p40 Ex 5.3 Eg 2) 12x – y = 21 3x + 7y = 27 Eg 3) 5x + 2y = -2 4x + 3y = 4 Substitution method If one equation is written y = or x = the substitution method is best y = 4x + 5 y = -2x +6 2y + 3x – 18 = 0 Theta y + 7x – 27 = 0 y = 4x – 12 y = 3x p40 Ex 5.4 Equations Steps: y = 4x + 5 1) Substitute the first equation in place of y in 2nd y + 7x – 27 = 0 2) Select one variable to eliminate 4x + 5 + 7x – 27 = 0 11x – 22 = 0 11x = 22 x=2 Simplify and solve (brackets may be needed) y = 4x + 5 y=4x2+5 y=8+5 y = 13 3) Substitute this variable back into one equation to get the other variable y + 7x – 27 = 0 13 + 7x2 – 27 = 0? 13 + 14 – 27 = 0 27-27 = 0 Yes! 4) Substitute both solutions into the other original equation to check answer is correct One solution Other solution Solution: x = 2 and y = 13 or coordinate (2, 13) Applications The entrance fees to the zoo are $12 for adults, and $8 for children One day 16 people go to the zoo, paying a total of $164 Applications p41 Ex 5.5 How many adults and children went to the zoo? Define variables x = adults y = children Form equations Total people x + y = 16 Total cost for adults = 12x and 8y for children Total combined cost = 12x + 8y = $164 Equations x + y = 16 12x + 8y = 164 Solution 9 adults 7 children Solve 8x + 8y = 128 12x + 8y = 164 -4x = -36 One solution x=9 Substitute x + y = 16 9 + y = 16 so y = 7 Check 12x + 8y = 164 12x9 + 8x7 = 164? 108 + 56 = 164 Yes! So solution is correct