Adding two equations gives a valid new equation. If one variable has opposite coefficients, it vanishes.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
When it is immediate
For 3x + y = 10 and 2x − y = 5, the y terms cancel on addition without any preparation.
Making it work
Multiply one or both equations by a constant to create opposite coefficients. Multiplying an equation throughout never changes its solutions.
Choosing multipliers
Match the coefficients to their lowest common multiple. For 2x and 3x, scale to 6x and −6x.
Elimination or substitution
Substitution suits a coefficient of 1. Elimination suits equations already in standard form.
The special cases return
If both variables vanish leaving something false, there is no solution. Leaving something true means infinitely many.
Add to remove a variable
If the coefficients of one variable are opposites, adding the equations eliminates it. The whole method is arranging for that cancellation and then solving what remains.
Scale first when needed
Multiply one or both equations so a variable's coefficients match or oppose. Multiplying an equation by a number keeps it true, provided every term on both sides is multiplied.
Subtracting means subtracting everything
When coefficients match rather than oppose, subtract — and subtract every term, including the constants. A dropped sign in that subtraction is the standard elimination error.
Choosing between the methods
Substitution suits systems with an isolated variable; elimination suits standard form with convenient coefficients. Both always work, so choose whichever the numbers make easier.
Step 2: Try It Yourself
Tap and try it out.
The slope is -2: for every 1 across, the line goes 2 down.
Step 3: Watch an Example
One step at a time.
Watch Diego Eliminate
Diego solves 2x + 3y = 12 and 4x − 3y = 6.
- Step 1
The y coefficients are 3 and −3, which are already opposites.
Step 4: Your Turn
Practice makes it stick.
The Cancellation
Problem 1 of 2
2x + 3y = 12 and 4x − 3y = 6. What is x?
The Second Value
Problem 2 of 2
Same system. What is y?
Cancel a Variable
1 of 8
x + y = 10 and x − y = 4. What is x?
2 of 8
Same system. What is y?
3 of 8
3x + y = 10 and 2x − y = 5. What is x?
4 of 8
2x + 2y = 14 and x − y = 1. What is x?
5 of 8
To eliminate from 2x and 3x, what common multiple do you scale to?
6 of 8
Does multiplying an equation throughout change its solutions? 1 yes, 0 no.
7 of 8
Sort each system by the better first method.
Tap something to move it.
- Empty
- Empty
8 of 8
4x + y = 14 and 2x − y = 4. What is x?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x + y = 12 and x − y = 2. What is x?
Question 2 of 2
Why may an equation be multiplied by a constant?
What You Learned
- Adding two equations eliminates a variable when its coefficients are opposites.
- Scale one or both equations to create those opposites.
- Substitution suits a coefficient of 1; elimination suits standard form.