Solve Systems by Elimination

Students solve systems by adding or subtracting equations to eliminate one variable, sometimes first multiplying one or both equations by a constant. They interpret the result when all variables cancel as indicating no solution or infinite solutions.

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Grade 8 · Systems of Linear Equations · question 1 of 3

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2x + 2y = 10, 2x + 4y = 14, y =

How it is taught

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A system is two equations that must both be true.

Elimination cancels one variable so you can solve.

Read it: Solving by elimination

2x + 3y = 18 and 2x + y = 10 both have 2x. Take the second from the first: 2x − 2x is 0, 3y − y = 2y, and 18 − 10 = 8. So 2y = 8 and y = 4.

Find a term both equations share, like 2x in each.
Take one equation from the other, side by side, so that term is gone.
Solve for the letter that is left, then put it back to find the other.

Equal amounts taken from equal amounts leave equal amounts, so the difference of the two equations is still true.

The 2x terms cancel: 2y = 8 has only y in it, so y = 4.

Put y = 4 into 2x + 4 = 10: 2x = 6, so x = 3.

Elimination: line up a matching term, take one equation from the other, and solve what is left.

Adding the equations when the matching terms are equal: 4x + 4y = 28.

Equal terms are taken away, not added: 2x − 2x = 0, which leaves 2y = 8.

8x + y = 19, 8x + 4y = 52, y = ___ — 11. The 8x terms match, so take one from the other: 3y = 33, and y = 11.

2x + 3y = 12 and 2x − y = 4

Today we'll solve a system of two equations using elimination. Our goal: find the x and y that make BOTH equations true.

Here are our two equations. Notice both have 2x — that's our chance to eliminate x.

Look! The x-terms match: 2x in the top, 2x in the bottom. If we subtract, the x's will cancel.

Let's subtract the bottom equation from the top equation, term by term.

Two x minus two x equals zero. The x is eliminated!

Three y minus negative y is three y plus y, which is four y.

Twelve minus four is eight. So four y equals eight.

Divide both sides by four. y equals two.

Grade 8 · Cancel a variable by subtracting equations

Now plug y equals two back into the first equation: 2x plus 3 times 2 equals 12.

Three times two is six, so 2x plus six equals twelve.

Subtract six from both sides, then divide by two. x equals three.

The solution is x equals three, y equals two. One point makes both equations true!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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