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