System of Equations and Elimination Section 4.3

Overview Solving by the Elimination Method Problem solving

Solving by the Elimination Method Graphing can be imprecise for solving systems Substitution method from previous section is precise but sometimes difficult to use For Example, to solve the system: 2x + 3y = 13 4x – 3y = 17 we would need to solve for a variable in one of the equations

Solving by the Elimination Method Elimination method uses addition principle a = b is equivalent to a + c = b + c If a = b and c = d, then a + c = b + d

Solving by the Elimination Method 2x + 3y = 13 4x – 3y = 17 (2x + 3y) + (17) = (13) + (17) (2x + 3y) + (4x – 3y) = (13) + (4x – 3y) (2x + 3y) + (4x – 3y) = (13) + (17) 2x + 4x + 3y – 3y = x + 0y = 30

Solving by the Elimination Method 2x + 3y = 13 4x – 3y = 17 6x + 0y = 30 6x = 30 x = 5 2x + 3y = 13 2(5) + 3y = y = 13 3y = 3 y = 1 (5, 1) Check answer!

Solving by the Elimination Method What if no terms are opposite of each other Example. Solve: 2x + 3y = 8 x + 3y = 7 Multiply one of the equations by –1 on both sides 2x + 3y = 8 -x - 3y = -7 x = 1 x + 3y = y = 7 3y = 6 y = 2 (1,2) Check Answer

Solving by the Elimination Method What if no terms have coefficients that are opposites of each other Example. Solve: 3x + 6y = -6 5x - 2y = 14 3x + 6y = -6 15x – 6y = 42 18x = 36 x = 2 (3)(2) + 6y = y = -6 6y = -12 y = -2 (2,-2) Check Answer

Solving by the Elimination Method Hint: Rearrange equations into Ax + By = C Form Solve: 3y x = 0 5x = 7 - 4y 2x + 3y = -1 5x + 4y = 7 Could multiply first equation by x – 7.5y = 2.5 5x + 4y = 7 0x – 3.5y = 9.5 y = (9.5)/(-3.5) y = Plug y = into original equation to find x ( , ) (3.57, 2.71)

Solving by the Elimination Method Could multiply first equation by -5/2 Solve: 3y x = 0 5x = 7 - 4y 2x + 3y = -1 5x + 4y = 7 -5x – 15/2y = 5/2 5x + 4y = 7 -5x – 15/2y = 5/2 5x + 8/2y = 14/2 0x – 7/2y = 19/2 (-2/7)(-7/2y) = (19/2)(-2/7) y = -19/7 2x + 3(-19/7) = -1 2x + (-57/7) = -1 2x = 50/7 x = 25/7 (25/7, -19/7) Check Answer

Solving by the Elimination Method 3y x = 0 5x = 7 - 4y 2x + 3y = -1 5x + 4y = 7 10x + 15y = x – 8y = -14 7y = -19 y = -19/7 2x + 3y = -1 2x + 3(-19/7) = -1 2x – 57/7 = -1 2x = /7 2x = -7/7 + 57/7 2x = 50/7 x = 25/7 (25/7, -19/7) Check answer

Solving by the Elimination Method NOTE: Could have also found a common multiple for coefficients of y variable 3y x = 0 5x = 7 - 4y 2x + 3y = -1 5x + 4y = 7 -8x - 12y = 4 15x + 12y = 21 7x = 25 x = 25/7 2x + 3y = -1 2(25/7) + 3y = -1 50/7 + 3y = -1 3y = -1 – 50/7 3y = -57/7 y = -19/7 (25/7, -19/7) Check answer

Solving by the Elimination Method Many ways to solve this system: 3y x = 0 5x = 7 - 4y 2x + 3y = -1 5x + 4y = 7 Multiple first equation by -2.5 or -5/2 Multiply second equation by or -3/4 Multiply first by -5 and second by 2 (or first by 5, second by -2) Multiply first by 4, second by -3 (or first by -4, second by3)

Solving by the Elimination Method DON”T DO THIS 3y x = 0 5x = 7 - 4y 2x + 3y = -1 5x + 4y = 7 2x + 3y = -1 5x + 4y –y = -1 –y 2x + 3y = -1 5x + 3y = -1 – y 2x + 3y = -1 -5x – 3y = 1 + y -3x = y

Solving by the Elimination Method Solve: y – 3x = 2 y – 3x = -1 Solve: 2x + 3y = 6 -8x – 12y = -24 Answer: {(x,y)|2x + 3y = 6} Or {(x,y)|-8x – 12y = -24}

Solving by the Elimination Method What if we have fractions or decimals? Multiply to clear them Solve: (1/2)x + (3/4)y = 2 x + 3y = 7 Multiply both sides of equation with Least Common Denominator (LCD), which is 4 2x + 3y = 8 x + 3y = 7 2x + 3y = 8 -x – 3y = -7 x = y = 7 3y = 6 y = 2 (1,2) Check answer

Problem Solving A calling card company offers two prepaid cards for domestic calls. The Liberty Prepaid Card and the USA Calling Card. The Liberty Card has a 50 cent connection fee per call and a 1 cent per minute rate. The USA Card has a 13.6 cent connection fee per call and a 2.2 cent per minute rate. For what length of call will the costs be the same?