How to Solve a System of Equations (Elimination Method)

A system of equations is two or more equations that must be true at the same time. The solution is the pair (x, y) that makes both true, which is where their graphs cross.

Try it with your own problem ↓

The steps

Line up the equations so the x terms, y terms and numbers are in columns.
Pick a variable to eliminate. If its coefficients are opposites (like +y and −y), add the equations. If they are the same, subtract them. Otherwise multiply one or both equations first so they match.
Add (or subtract) the equations. One variable disappears.
Solve the single equation that is left.
Put that value into either original equation to find the other variable, then check in both.

A worked example

Solve 2x + y = 7 and x − y = 2.

The y terms are +y and −y, which are opposites, so add the equations.
(2x + y) + (x − y) = 7 + 2, which gives 3x = 9.
Divide by 3: x = 3.
Put x = 3 into x − y = 2: 3 − y = 2, so y = 1.
Check both: 2(3) + 1 = 7 ✓ and 3 − 1 = 2 ✓.

x = 3, y = 1, or the point (3, 1).

Try it with your own problem

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Work it out: 2x + y = 7; x - y = 2Opens The Balance Scale with this example

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Common mistakes

Adding when you should subtract. Add when the coefficients are opposites. Subtract when they are equal.
Multiplying only one term. When you scale an equation, multiply every term, including the number on the right.
Finding x and stopping. A solution needs both x and y.
Checking in only one equation. The pair must work in both.

Questions people ask

What is the substitution method?

Solve one equation for a variable, such as y = 2x − 7, then put that expression in place of the variable in the other equation. It's best when a variable already has a coefficient of 1 or is alone.

What if there is no solution?

If you get something false, like 0 = 5, the lines are parallel and never meet. If you get something always true, like 0 = 0, they are the same line and there are infinitely many solutions.

Which method should I use?

Any works. Elimination is quick when the coefficients line up, and graphing helps you see what's going on.

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