Over the years I have seen that students really struggle with solving systems of equations by elimination. When teaching I try to find a process that will be consistent so students can repeat the same steps and eventually remember how to solve a particular problem. There are definitely faster ways to solve the systems than I teach them but this process works for all the problems if they are in standard form.
I have students focus only on eliminating the x's even if the y's look easier. Once they understand the process I then tell them about finding shortcuts. So if we focus on eliminating the x's first I tell them the coefficient has to have the exact same number but different signs so that the x's cancel out to zero and disappear. We practice flipping the coefficients and then multiplying to essential have the same coefficient for the x's. Then I ask them if they already have different signs. If they do then we go into the elimination step. If they do not I tell them to ONLY multiply the TOP equation by a negative. Before we go into the entire process I practice just determining what to multiply each equation by to solve. So I created a card sort activity to practice that. Remember that I am focusing on the x's. If you do not do it this way then you will probably find these resources need to be modified to fit your needs.
Here are the notes I start with to teach this topic. We focus on the rule part at the top for a little while then finally move to actually solving the equations at the bottom once I feel like they understand what we need to multiply by to eliminate one of the variables.
Here is the card sort activity that I use to help reinforce what they must multiply by to have the x's eliminate.
You may notice that many of the problems have the same numbers in the front of each equation. I purposely did that because I wanted to force the students to really have to think about what they needed to multiply by and not just match up the numbers. Most of my card sorts I try to do this. I find similar problems that have only a few changes so they really have to focus on the rules and steps of a concept.
Like factoring trinomials for example, I try to make sure that other trinomials share a common factor and also make sure 3 or 4 trinomials all start with the same coefficient so it isn't easy to just match up by multiplying the first two parts of each factor.
Here is the dry erase template that I use to practice solving systems of linear equations by elimination.
***Update*** I totally forgot about the truths & lies activity that I made for elimination. I love this activity because it focuses on students having to find mistakes and fix those problems.
Hope you guys find these resources helpful. If you would like to download the files you can find them below.
Notes for Solving Systems of Equations by Elimination
Card Sort for Multiplication Process of Elimination
Dry Erase Template for Solving Systems of Equations by Elimination
Truths & Lies Activity for Solving Systems of Equations by Elimination