Presentation on theme: "Solving Systems of Linear Equations By Elimination"— Presentation transcript:
1 Solving Systems of Linear Equations By Elimination
2 What is Elimination? To eliminate means to get rid of or remove. You solve equations by eliminating one of the variables (x or y) using addition or subtraction.
3 Example 1 Solve the following system of linear equations by elimination.2x – 3y = 155x + 3y = 27(1)(2)Add equation (1) to equation (2)7x + 0y = 427x = 42 By eliminating y, we can now solve for xx = 6
4 Example 1 Substitute x= 6 into equation (1) to solve for y Check your solution x = 6 and y = -1 in equation (2)2x – 3y = 155x + 3y = 272(6) – 3y = 155(6) + 3(-1) = 2730 – 3 = 2712 – 3y = 1527 = 27– 3y = 15 – 12LS = RS– 3y = 3y = -1Therefore, the solution set = (6,-1)
8 If you have noticed in the last few examples that to eliminate a variable, it’s coefficients must have a sum or difference of zero.Sometimes you may need to multiply one or both of the equations by a nonzero number first so that you can then add or subtract the equations to eliminate one of the variables.2x + 5y = 17 7x + 2y = 10 2x + 5y = -226x – 5y = x + y = x + 3y = 22We can add these two equations together to eliminate the x variable.We can add these two equations together to eliminate the y variable.What are we going to do with these equations, can’t eliminate a variable the way they are written?
9 Multiplying One Equation Solve by Elimination2x + 5y = -2210x + 3y = 222x + 5y = (2x + 5y = -22) x y = -11010x + 3y = x + 3y = (10x + 3y = 22)y = -132y = -6
10 Step y = -6 Solve for the eliminated variable using either of the original equations x + 5y = -22 Choose the first equation. 2x + 5(-6) = -22 Substitute -6 for y. 2x – 30 = Solve for x x = x = 4 The solution is (4, -6).
11 Solve by elimination. -2x + 5y = -32 7x – 5y = x – 3y = 61 2x + y = x – 10y = -25 4x + 40y = 20
14 Multiplying Both Equations To eliminate a variable, you may need to multiply both equations in a system by a nonzero number. Multiply each equation by values such that when you write equivalent equations, you can then add or subtract to eliminate a variable.4x + 2y = 147x + 3y = -8In these two equations you cannot use graphing or substitution very easily. However ever if we multiply the first equation by 3 and the second by 2, we can eliminate the y variable.Find the least common multiple LCM of the coefficients of one variable, since working with smaller numbers tends to reduce the likelihood of errors.4 x 7 = 282 x 3 = 6
15 4x + 2y = 14 3(4x + 2y = 14) x + 6y = 42 7x – 3y = (7x – 3y = -8) x – 6y = x = x = x = 1 Solve for the eliminated variable y using either of the original equations. 4x + 2y = 14 4(1) + 2y = y = y = y = 5 The solution is (1, 5).