Elimination Method (Level 2) page 2 of 5
The Elimination Method with Multiplication
Sometimes when you add two equations together, none of the variables cancel out. If this happens, it doesn't mean you can't use the elimination method. It just means that you have to go back and "fix" the system so that one of the variables will cancel.
Think about what needs to happen for a variable to cancel. When you add the terms together, they need to add up to 0. In order for this to happen, the terms need to be opposites. For example, a 2x and a -2x will cancel. A 5y and a -5y will cancel.
If you don't have terms that are opposites, you can use multiplication to create a new system. To do this, you can multiply an entire equation by the same number. Remember, it's important to do the same thing to both sides of an equation to keep things balanced.
In the problem we looked at earlier, we had an x in the top equation and a 3x in the bottom. We need the x-terms to be opposites to cancel. This means we need a 3x and a -3x. To create a -3x term, we can multiply the entire top equation by -3.
Be careful on this step. The most common mistake is for students to multiply part of the equation by the constant, but not the whole thing. We need to make sure we multiply each term by -3 for this to work. Now that we've "fixed" the equation, the x's will cancel out when we add the two equations together. Once you get a variable to cancel, the rest of the problem is just the problems in the Level 1 Elimination lesson.
Here's a summary of the steps: