3.2.1 – Solving Systems by Combinations

We have addressed the case of using substitution with linear systems When would substitution not be easy to use?

Combinations Similar to substitution, we can use a new method when solving for a specific variable may not be easy Fractions Multi-step Odd Numbers

In order to use combinations, our goal is the following; “Knock out” or eliminate one variable. Solve for the remaining. Then, similar to substitution, go back and find the other missing variable

How to use To use the combination, or knock-out method, we do the following 1) Find the variable with the same coefficient in both equations; multiply to get the same coefficient if necessary 2) Add or subtract down, make sure one variable is eliminated 3) solve for the remaining variable 4) Go back to one of the original equations, and solve for the remaining variable 5) Check final solutions

To help, it’s generally easiest to line the equations up as if you were doing addition or subtraction like you first learned Add = if signs are opposite Subtract = if signs are same

Example. Solve the following system. 4x – 6y = 24 4x – 5y = 8

Example. Solve the following system. 2x – 8y = 10 -2x – y = -1

Example. Solve the following system. 3x – y = -3 x + y = 3

Multiplying As mentioned, sometimes the coefficients may not be the same Allowed to multiply one, or both equations, by a number to get the same coefficients for one of the variables Make sure to multiple every term!

Example. Solve the following system. 3x + 2y = -2 x – y = 11 Which variable should we try to cancel?

Example. Solve the following system. 5x – 2y = -2 3x + 5y = 36

Assignment Pg. 142 2, 4-6, 9-25 odd Pg. 143 38, 39