How to solve system of equations by elimination
- Markus Shobe
- 2 days ago
- 3 min read
By the time you finish reading this, you will know exactly how to solve a system of equations using elimination, why this method is so useful, and how it can help you feel more confident in math class and on tests. You will also see a clear example that makes the whole process feel manageable instead of confusing.
Solving systems of equations by elimination is a skill that shows up in middle school, high school, and beyond. It is especially common in Algebra 1 and Algebra 2. Many students struggle with it at first because there are multiple steps, but once you understand the goal of each step, it becomes very logical.
What is solving systems of equations by elimination?
A system of equations is just two equations with the same two variables, usually x and y. The goal is to find the one pair of values that makes both equations true at the same time.
The elimination method works by removing, or eliminating, one variable so you can solve for the other. After that, you plug the value back in to find the second variable. That is it. No magic, just organized steps.
Why elimination is such a powerful method
Elimination is great because it works even when graphing is messy and substitution feels too complicated. If the equations are lined up nicely, elimination is often the fastest and cleanest method.
Students gain a lot from learning elimination because it builds strong algebra skills like combining like terms, working with negatives, and staying organized. These skills show up again in harder math, so mastering elimination now makes future topics easier.
A step by step example
Let’s look at a simple system of equations.
Equation 1
x + y = 10
Equation 2
x − y = 4
Step 1. Line up the equations
Notice how x and y are already lined up. This makes elimination easier.
Step 2. Choose a variable to eliminate
If we add the two equations together, the y values will cancel out.
x + y = 10
x − y = 4
Add them together.
2x = 14
Step 3. Solve for x
2x = 14
x = 7
Step 4. Plug x back into one equation
Now put x = 7 into either equation.
x + y = 10
7 + y = 10
y = 3
Step 5. Write the solution
The solution to the system is x = 7 and y = 3.
This means the point (7, 3) makes both equations true.
What if the variables do not cancel right away?
Sometimes the variables do not cancel immediately. That is normal. In those cases, you multiply one or both equations so that one variable has the same number but opposite signs.
For example, if you had
2x + y = 8
3x − y = 7
The y values already have opposite signs, so adding still works. If they did not, you would multiply one equation to make them match. This part takes practice, but once students see a few examples, it clicks.
Common mistakes students make
One common mistake is forgetting to change every term when multiplying an equation. Another is mixing up signs when adding or subtracting equations. These are not big problems, but they can lead to wrong answers if not caught.
This is where guided practice really helps. Having someone check your steps and explain where things went off track can make a huge difference in confidence.
How I help students learn elimination
When I tutor students, I slow the process down and focus on why each step works. We do examples together, talk through mistakes, and build a simple routine they can follow every time. My goal is not just to get the right answer, but to help students feel in control of the math.
Once elimination makes sense, students often say systems of equations stop being scary and start feeling like a puzzle they know how to solve.
If you want your learner to feel more confident with systems of equations and other algebra topics, claim your first session 50% off with Markus at Precision Math Tutoring at this page
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