Presentation on theme: "Solving Systems of Linear Equations By Elimination"— Presentation transcript:
1Solving Systems of Linear Equations By Elimination
2What 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.
3Example 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
4Example 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)
8If 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?
9Multiplying 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
10Step 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).
11Solve by elimination. -2x + 5y = -32 7x – 5y = x – 3y = 61 2x + y = x – 10y = -25 4x + 40y = 20
14Multiplying 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
154x + 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).