Elimination Method
Elimination solves a system of equations by combining the two equations so one variable cancels out. If one equation has and the other has , adding the equations makes the terms disappear — and you are left with a one-variable equation you already know how to solve.
This method shines when both equations are written in the form $ax + by = c$. Instead of solving for a variable first (like substitution), you line the equations up and let the addition do the work.
The idea: make a variable cancel
Stack the two equations so the terms, terms, and constants line up in columns. Then add the equations top to bottom: terms with terms, terms with terms, right sides with right sides.
If one variable has opposite coefficients — like and , or and — those terms sum to zero and vanish. What remains is a single equation in a single variable.
Adding the same amount to both sides of an equation keeps it true, and each equation says its left side equals its right side. That is why adding two true equations produces another true equation.
Add or subtract?
Add when the coefficients of one variable are opposites: and cancel when added.
Subtract when the coefficients of one variable are identical: if both equations have , subtracting one equation from the other cancels the terms.
Many students avoid subtracting because it is easy to drop a negative sign. A safe alternative: multiply every term of one equation by first, then add. Same result, fewer sign mistakes.
Finish with substitution
Eliminating a variable only gets you half the answer — one coordinate. Substitute that value into either original equation to find the other variable, then write the solution as an ordered pair .
Check the pair in the equation you did not use for the substitution step. If it works there too, the solution is right.
Worked examples
Example 1: opposite coefficients, add
Solve the system: and .
Answer:
Example 2: matching coefficients, subtract
Solve the system: and .
Answer:
Example 3: eliminate y instead
Solve the system: and .
Answer:
Try one yourself
Common questions
How do I know which variable to eliminate?
Look for the variable whose coefficients are already opposites or already equal — that one cancels with a single add or subtract. If neither variable qualifies, you will need to multiply an equation first, which is its own skill: elimination with multiplication.
When should I use elimination instead of substitution?
Use elimination when both equations are in $ax + by = c$ form, especially when no variable has a coefficient of . Use substitution when one equation is already solved for a variable, like . Both methods always give the same answer.
What if both variables cancel at the same time?
Then the variables are gone and you are left comparing two numbers. If the statement is false, like , the system has no solution — the lines are parallel. If it is true, like , the equations are the same line and there are infinitely many solutions.
Does it matter which equation I substitute back into?
No — either original equation gives the same value. Pick whichever has smaller numbers, and use the other one as your final check.
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