Solving Systems of Linear Equations By Elimination

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.

Example 1 Solve the following system of linear equations by elimination. 2x – 3y = 15 5x + 3y = 27 (1) (2) Add equation (1) to equation (2) 7x + 0y = 42  7x = 42  By eliminating y, we can now solve for x x = 6

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 = 15 5x + 3y = 27 2(6) – 3y = 15 5(6) + 3(-1) = 27 30 – 3 = 27 12 – 3y = 15 27 = 27 – 3y = 15 – 12 LS = RS – 3y = 3 y = -1 Therefore, the solution set = (6,-1)

You Try 5x – 6y = -32 3x + 6y = 48

One More 7x + 2y = −19 −x + 2y = 21

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 = -22 6x – 5y = -19 -7x + y = -16 10x + 3y = 22 We 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?

Multiplying One Equation Solve by Elimination 2x + 5y = -22 10x + 3y = 22 2x + 5y = -22 5(2x + 5y = -22) 10x + 25y = -110 10x + 3y = 22 10x + 3y = 22 - (10x + 3y = 22) 0 + 22y = -132 y = -6

Step 2 y = -6 Solve for the eliminated variable using either of the original equations. 2x + 5y = -22 Choose the first equation. 2x + 5(-6) = -22 Substitute -6 for y. 2x – 30 = -22 Solve for x. 2x = 8 x = 4 The solution is (4, -6).

Solve by elimination. -2x + 5y = -32 7x – 5y = 17 2x – 3y = 61 2x + y = -7 3x – 10y = -25 4x + 40y = 20

5x + 4y = -28 3x + 10y = -13

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 = 14 7x + 3y = -8 In 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 = 28 2 x 3 = 6

4x + 2y = 14 3(4x + 2y = 14) 12x + 6y = 42 7x – 3y = - 8 2(7x – 3y = -8) 14x – 6y = -16 26x + 0 = 26 26x = 26 x = 1 Solve for the eliminated variable y using either of the original equations. 4x + 2y = 14 4(1) + 2y = 14 4 + 2y = 14 2y = 10 y = 5 The solution is (1, 5).

Solve by your method of choice: 1) 2x + 5y = 17 2) 7x + 2y = 10 Closure Solve by your method of choice: 1) 2x + 5y = 17 2) 7x + 2y = 10 6x + 5y = -9 -7x + y = -16 3) 2x – 3y = 61 4) 24x + 2y = 52 2x + y = -7 6x – 3y = -36 5) y = 2x 6) 9x + 5y = 34 y = x – 1 8x – 2y = -2 1. (1, 3) 2. (2, -2) 3. (5, -17) 4. (1, 14) 5. (-1, -2) 6. (1, 5)

Solve Systems of Equation by Elimination in Real World Context

Two groups of people went to see Guardians of the Galaxy in IMAX 3‐D Two groups of people went to see Guardians of the Galaxy in IMAX 3‐D. The first group spent $73.50 on two adult and three children tickets. The other group spent $109.50 on five adult and two children tickets. What is the cost for each type of ticket?

For dinner, Pat had a double cheeseburger and two medium fries totaling 1200 calories. Matt has two double cheeseburgers and one medium fries totaling 1260 calories. How many calories are in one double cheeseburger? One order of medium fries?

For breakfast, Bill had a bacon, egg, and cheese biscuit and two hotcakes totaling 660 calories. Phil had a bacon, egg, and cheese biscuit and three hotcakes totaling 780 calories. How many calories are in one hotcake? One bacon, egg, and cheese biscuit?

On December 9th, Trevor Ariza, of the Houston Rockets, scored 34 points. What is odd about this is that he scored all 34 points only on 2‐pointers and 3‐pointers. He made a total of 15 shots. How many 3‐pointers did he make?

At McDonalds, a cheeseburger has 200 fewer calories than a large fries At McDonalds, a cheeseburger has 200 fewer calories than a large fries. Two cheeseburgers and a large fries have 1100 calories. How many calories are in each item?

At Billy’s preschool, they have a total of 25 bicycles and tricycles At Billy’s preschool, they have a total of 25 bicycles and tricycles. Among them all, there are 57 wheels. How many of each are there?