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Review of the Elimination MethodIf you're new to elimination, you may want to check out the Elimination Level 1 lesson first for some simpler problems.
The goal with the elimination method is to get one of the variables to cancel out when you add the two equations together. In the Level 1 lesson, either the x's or the y's canceled when we added the equations. For example, if you add the two equations below the x terms will cancel because you have a positive 3x and a negative 3x. In the system above, the x's cancel and we're left with an equation that's pretty easy to solve for y. Unfortunately, this doesn't happen every time. You might get a system that's a little more complicated. Take a look at the system below. If we add the two equations, we still have both variables left. This is a problem we'll have to fix. Open the next tab to see how to fix this type of problem. The Elimination Method with MultiplicationSometimes 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 xterms 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: Example 1
Solve the system. 2x  5y = 6 and 3x + y = 26 Step 1: Use multiplication to create terms that are opposites. We can create opposites by multiplying the entire 2nd equation by 5. This will give us a 5y and a 5y that will cancel. Step 2: Add the equations vertically. Now that we've created terms that are opposites, one of the variables will cancel out when we add the equations. Step 3: Solve for the first variable. The y terms canceled, so we can solve the equation that's left. Step 4: Solve for the second variable. Go back to one of the original equations (you can choose either one) and plug in 8 for x. Step 5: Write answer as ordered pair. Make sure to put the variables in alphabetical order when you write the ordered pair. Step 6: Check your answer. Use the original equation that you didn't use earlier. If you use the same equation twice, it will not find a mistake. Example 2 Solve the system. 2x + 5y = 1 and 3x  2y = 11 Step 1: Use multiplication to create terms that are opposites. This time it's a little trickier to create opposites. We will need to multiply BOTH equations by different numbers to end up with terms that are opposites. You can choose to get the x's or y's to cancel, it doesn't matter. Let's choose the x's to cancel. Right now we have a 2x and a 3x. If we multiply the first equation by 3 and the second equation by 2, we'll get 6x and 6x. It's somewhat like the process of finding a common denominator when you're adding fractions. Sometimes you need to multiply both equations to get terms that will cancel. Once you get terms to cancel, the rest of the problem is pretty straightforward. Solve for each variable. Write answer as an ordered pair and check by plugging the values into an original equation. YouTube blocked? You can also see the video here. Powered by Interact 
