# What is the Elimination Method?

To solve a system of equations using elimination or addition, begin by rewriting both equations in standard form Ax+By=C. Check to see if the coefficients of one pair of like variables add to zero. If not, multiply one or both of the equations by a non-zero number to make one set of like variables add to zero. Add the two equations to solve for one of the variables. Substitute this value into one of the original equations to solve for the other variable. Check your work by substituting into the other equation.

## The Elimination Method

In this video, we’re going to look at the elimination method for solving systems of equations. Let’s take a look at an example problem. Let’s say we have the equations 5x plus 6y equals 24, and 3x minus 2y equals 4. We have these two equations.

**The goal of the elimination method is to eliminate one of the two variables from these equations so we can solve for the other one.** We do that by getting the coefficients of one of those two variables to be the same as the coefficient of that variable in the other equation.

For instance, in this case, we’re going to use y, because 6 is a multiple of 2. To get the coefficient of y in the second equation to equal 6, all we have to do is multiply each term in the equation by 3. This equation here now becomes 9x minus 6y equals 12. This equation up here retains its original values.

**Now we have these two equations, which are equivalent to these two equations.** With these equations in these forms, we can add the two equations together, and thereby eliminate y. If we add 5x and 9x, we get 14x. If we add 6x and -6x, we get zero here. That’s empty. 24 and 12 are 36.

We have the equation 14x equals 36, which we can easily solve as x equals 36/14, or divide top and bottom by 2. That’s the same as 18/7. We have our value for x now. Now, we can substitute this value back into one of these equations to solve for y. We’ll plug it back into this second equation, because the numbers are smaller.

We can solve for y as 3 times 18/7 minus 2y equals 4. The first thing we want to do is get y by itself. We’ll add 2y to both sides and subtract 4 from both sides. That will eliminate that 4 and that y. We’ll have 3 times 18/7 minus 4 is equal to 2y. 3 times 18/7. 3 times 18 is 54.

54/7 minus 4 is equivalent to 28/7. 54/7 minus 28/7 equals 2y, and 54 minus 28 is 26. We have 26/7 is equal to 2y, or y equals 13/7. That is the elimination method for solving a system of equations. We manipulate one or both of the two equations, such that we can get two of the coefficients to be the same value.

Then, we either add or subtract the two equations. If this had been plus 6y instead of minus 6y, we would have subtracted the two equations. Because we had a plus and minus, we added, and that got us this equation where we just have x without y. We were able to solve for x, plug x back into one of these equations, and solve for y. That’s the elimination method.