TL;DR:
Elimination makes one variable cancel by scaling the equations and adding or subtracting them.
Adding equal things to equal things keeps both sides equal, so the new equation is still true.
Always check the pair in both original equations.
A café sells two items and you know two facts about one day's sales. Each fact is an equation with two unknowns, and neither alone can be solved. Together they can. A system of equations is the pair, and elimination is a way to combine them so that one unknown disappears.
The move that keeps equations true
If A = B and C = D, then A + C = B + D. You are adding equal amounts to both sides, so the equation stays true. Elimination uses this on purpose: it adds two equations so that the terms in one variable cancel, leaving a single equation you can solve.
You may also multiply a whole equation by a non-zero number. Every term changes, both sides stay equal, and the solution does not move. That is how you make the coefficients line up before adding.
Worked example: 2x + 3y = 12 and x − y = 1
The goal is to cancel y. In the first equation y has coefficient 3, in the second −1. If we multiply the second equation by 3, its y coefficient becomes −3, the opposite of +3.
Label the equations: (1)
2x + 3y = 12, (2)x − y = 1.Multiply (2) by 3:
3x − 3y = 3. Call it (3).Add (1) and (3):
2x + 3x + 3y − 3y = 12 + 3, so5x = 15.Solve:
x = 3.Put
x = 3into the simpler equation (2):3 − y = 1, soy = 2.
The solution is x = 3, y = 2, which is the point (3, 2) where the two lines cross.
Check in both equations
Checking in only one equation is not enough, because a mistake can leave you with a point on one line but not the other. Equation (1): 2(3) + 3(2) = 6 + 6 = 12. Correct. Equation (2): 3 − 2 = 1. Correct. Both hold, so the pair is the answer.
A harder pair, scaling both equations
Sometimes neither coefficient is a simple multiple of the other. Take (1) 3x + 4y = 10 and (2) 5x + 6y = 14. To cancel x, aim for the least common multiple of 3 and 5, which is 15.
Multiply (1) by 5:
15x + 20y = 50.Multiply (2) by 3:
15x + 18y = 42.Subtract the second from the first:
2y = 8, soy = 4.Back-substitute into (1):
3x + 16 = 10, so3x = −6andx = −2.
Check in (2), the equation not used for back-substitution: 5(−2) + 6(4) = −10 + 24 = 14. Correct. Checking in the one you did not use is a good habit because it tests the work from a different angle.
A word problem with the same shape
Say two coffees and one muffin cost $11, and one coffee and three muffins cost $13 (made-up prices). Let c be a coffee and m a muffin. Then (1) 2c + m = 11 and (2) c + 3m = 13. Multiply (1) by 3: 6c + 3m = 33. Subtract (2): 5c = 20, so c = 4. Then m = 11 − 8 = 3. Check in (2): 4 + 3(3) = 13. Correct. A coffee is $4 and a muffin is $3.
Notice that the algebra never needed the story. The skill is turning two facts into two equations with clearly defined letters, then letting the method run.
When there is no single answer
Two special things can happen. If you add and everything cancels to a false statement such as 0 = 3, the lines are parallel and never meet, so there is no solution. For example, x + y = 2 and x + y = 5 cannot both be true. If everything cancels to a true statement such as 0 = 0, the equations describe the same line and there are infinitely many solutions, as with x + y = 2 and 2x + 2y = 4.
When substitution is easier
Use substitution when one equation already has a variable alone, such as y = 2x + 1. Then you can drop that expression into the other equation directly. Elimination is usually cleaner when both equations are in the form ax + by = c, as above. Both methods give the same answer, so choose the one that needs fewer steps.
Common mistakes
Multiplying only the left side by a number, and forgetting the right side.
Adding when the signs are the same, which doubles the term instead of cancelling it. If the coefficients match, subtract; if they are opposite, add.
Subtracting and mishandling the signs, such as
20y − 18ywritten as−2y. A careful way is to write the subtraction as adding the negated equation.Solving for one variable and forgetting the other.
Checking in only one equation.
Where you will use it next
Systems appear whenever two conditions must hold at once, such as a cost and a headcount. Each equation is a line, and the slope of each line is covered in slope as a rate of change. With three unknowns, the same scaling and adding idea carries on, and with many unknowns it becomes the matrix method used by computers.
Frequently asked questions
Does elimination always give the same answer as substitution?
Yes. They are two routes to the same crossing point, as long as you do not make an arithmetic slip.
What if the numbers come out as fractions?
That is allowed. Check by substituting the fractions back, or scale the equations differently so the arithmetic is cleaner.
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