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Mastering System Solutions: The Knock-Out Method for Linear Equations

This guide covers using the combination method, also known as the knock-out method, to solve systems of linear equations, especially when substitution isn't practical. The process involves eliminating one variable to find the other, making it a valuable technique for multi-step problems or those involving fractions. We’ll go through the steps for using this method, complete with examples and tips for ensuring accuracy. Practice assignments included to reinforce learning and mastery of solving systems of equations.

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Mastering System Solutions: The Knock-Out Method for Linear Equations

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  1. 3.2.1 – Solving Systems by Combinations

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

  3. 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

  4. 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

  5. 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

  6. 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

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

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

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

  10. 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!

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

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

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

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