At this point, students have already learned how to solve a system of equations by graphing and by substitution. We’ll introduce the elimination method first by reviewing distributive property and writing an equivalent equation. We’ll then go through examples as a class, adding an additional piece of difficulty with each example. Students will finish with a ByteLearn classwork assignment. You can expect this lesson to take one `45`-minute class period.
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Students will be able to solve a system of equations using the elimination method.
Give students a few minutes in the beginning of class to try out these two warm-up problems. The first example formally asks students to simplify using the distributive property. The second example asks students to multiply the equation by `4`.
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The key idea as you are going over these examples as a class is that we can multiply entire equations by a number, similar to how we use the distributive property! Be sure to note that the same rules apply: we need to be sure to multiply each term in the expression or equation by the factor.
After going over the second problem as a class, ask students if the equations `2x - 3y = -8` and `8x - 12y = -32` are equivalent. We want students to recognize that these are two different forms of the same linear equation. Explain that they are equivalent because what we multiplied by one one side of the equation, we also multiplied by on the other, keeping the equation balanced.
The first example allows us to add to eliminate a variable. Show students slide `2` and explain that in the elimination method, we combine the two equations in a way that allows us to eliminate one of the variables. Point out to students that the equations are stacked to line up the like terms. Ask students, “what would happen if we added `-10x` and `-4x`?”. Ask the same for `-5y` and `5y`. Students should be able to identify that `-5y` and `5y` would cancel out if added. So show students that in order to cancel out the `y` variable, we can add the equations. Once we find the value of `x`, remind students that we can use either equation to find the value of `y`. The work for how you can solve this problem on the board is shown below:
For this next example, ask students if one of the variables would cancel if we were to add the two equations. Hopefully, students will recognize that neither will cancel when added.
There are two ways to teach students to solve a problem like this:
I’ve taught this both ways and have found students prefer multiplying by `-1`, then adding. You can show students both methods and see which they prefer! Also point out to students that we eliminated the `x` in this example, but we eliminated the `y` in the previous example. Talk about how you can eliminate either variable depending on the problem and which coefficients are easier to eliminate.
For example `3`, neither variable’s coefficients are the same or opposite. Ask students to think back to the warm up and how we can write equivalent equations. Point out to students that the coefficients for `y` are `2` and `6`. Tell students to notice that `6` is a multiple of `2`. If we want to cancel the `y`-terms by adding the equations, we would need the coefficient in the first equation to be `-6`. Ask students, “what can we multiply the first equation by so that the coefficient for `y` becomes `-6`?” Students will likely be able to identify that we would multiply the equation by `-3`. Guide students as we write an equivalent equation after multiplying the first equation by `-3`. From there, you can let students take over, as they will already be becoming familiar with how to add the two equations to eliminate the `y`.
For the last type of example, things become a bit more challenging for students. Allow students a few minutes to discuss how they might be able to eliminate a variable in this system. You may have some students work together to figure out that you can multiply both equations by a factor. If no students bring up that idea, try asking, “in the last few examples, we multiplied one equation by a number so that we were able to cancel. Is it possible we can multiply both equations by different numbers to cancel a variable?”
Allowing students to come up with the numbers they want to multiply on their own is a great way to have the class recognize that there are multiple ways to solve the problem.
Now it’s time for some independent practice! You can assign a ByteLearn online practice to your class using the link below. Students will get immediate feedback and step-by-step help if they need it. Set a due date and allow students to finish the assignment for homework. Once complete, you’ll see detailed reports of students who may need additional support, students who are ready for a challenge, and other interesting insights!
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