Any Learning Objectives:
1. Students will acquire the concept of the linear combination method of solving systems of equations.
2. Students will accurately solve systems of equations using the linear combination method.
3. Students will verbalize how to choose between the different methods of solving systems of equations. Materials:
Attached flowchart handout Procedures:
Remind students of the material covered in the previous lesson. Give them a quick system of equations to solve with their partners by substitution.
Ask the students if they think this method is easier than graphing? Elicit a brief discussion of the relative merits of the substitution method over graphing. Also highlight (if students don’t bring it up), its drawbacks – namely, that it requires a lot of transforming of various equations and a lot of fractions. Give the following as an example: 5x + 3y = 27 7x – 3y = 45
Briefly, have the students explain how they would go about solving this using substitution. Demonstrate these transformations on the board, and show how there is a large amount of fraction usage. Wouldn’t it be nice if there were a more efficient way of solving this system of equations? Tell them that they will be learning a method of solving systems just such as these today -- and no fractions are necessary!
Remind them that it is always permissible to add the same thing to both sides of an equation – elicit that this is the Addition Axiom of Equality. Thus, if I take the first of my two equations, I can add the same thing to both sides, right? What if I were to add 45 to both sides? 5x + 3y = 27 5x + 3y +45 = 27 + 45
This is allowed, right? But since we already have an equation relating “45” and an algebraic expression, and since they are equal, can’t I replace “45” with what it is equal to (in this case, 7x- 3y)? What property allows me to do that? (Elicit – the Substitution Axiom of Equality.)
Thus, 5x + 3y +45 = 27 + 45 5x + 3y + (7x – 3y) = 27 + 45
Now, what do you notice about the “y” terms in my new equation? They cancel out! That will allow me to solve for x and then, alternatively to solve for y.
But, look what happens if I line them up! (Demonstrate the traditional method of “adding” the two equations together, and show that this is identical to the method used above.)
Provide another example for the students to work with their student pairs: -4x + 8y = -3.6 4x – 3y = 13.1
Have the class come together and report on the answer (4.7, 1.9) and their means of solving it.
Next, provide the following example for them to attempt: 3x + 5y = 17 2x + 3y = 11
Before allowing the student pairs to work, ask if they notice anything that would make this difficult? (Elicit that there are no coefficients that are equal.) Remind them that the Multiplication Axiom of Equality allows them to multiply both sides of an equation by anything. Thus, if you multiply the top equation by “2” and the bottom equation by “3”, you get the following system: 6x + 10y = 34 6x +9y = 33
Have the students then quickly solve this system of equations with their partners for the correct solution of (4, 1).
Briefly lead a discussion of which method they find easier (it will probably be linear combination). Lead them in a reflection on which method is easier in given situations. Distribute the attached flowchart for student reference. Attached Files: