1.6 - Solving Linear Systems: Elimination

• Recall: In 1.5, we solved a system of equations by substituting one equation into the other

• Now, in 1.6, we will solve a system of equations by eliminating/removing a variable

Everyday, Brenda bakes chocolate chip cookies and low-fat oatmeal cookies. She uses different amount of butter and oatmeal in each recipe. Brenda has 47 kg or butter and 140 kg of oatmeal

Chocolate Chip Low-Fat Oatmeal

13 kg butter 2 kg butter

8 kg oatmeal 29 kg oatmeal

Example #1

How many batches of each type of cookie can Brenda make using all the oatmeal and butter she has?

Chocolate Chip Low-Fat Oatmeal

13 kg butter 2 kg butter

8 kg oatmeal 29 kg oatmeal

Let r be the number of batches of chocolate chip cookies. Let s be the number of batches of low-fat cookies.1. 13r + 2s = 472. 8r + 29s = 140Let’s multiply “1.” by 8 and multiply “2.” by 13 – this will allow us to eliminate the ‘r’ values:1. x 8: 8(13r + 2s) = 8(47) 104r + 16s = 376 2. X 13: 13(8r + 29s) = 8(140) 104r + 377s = 1820

Example #1cont’d 104r + 16s = 376 - 104r + 377s = 1820 - 361s = -1444s = s = 4Plug s = 4 back into one of original equations to find ‘r’:13r + 2(4) = 4713r = 39r = 3

Therefore, Brenda can make three batches of chocolate chip cookies and four batches of low-fat cookies.

Example #2

Let x represent the distance that the submarine travelled on the surfaceLet y represent the distance that it travelled underwater.1. x + y = 2002.

During a training exercise, a submarine travelled 20 km/h on the surface and 10 km/h underwater. The submarine travelled 200 km in 12 hours. How far did the submarine travel underwater?

Example #2 cont’dMultiply “2.” by 20 so we can eliminate x:1. x + y = 2002. x 20: 20(x + 2y = 240

Use the new “2.” to eliminate ‘x’ from “1.”: x + 2y = 240 – x + y = 200 y = 40

Now find x:x + y = 200x = 200 – y

= 200 – 40x = 160

Therefore, the submarine travelled 40 km underwater.