Algebra · Step-by-step guide

The Elimination Method

Solve linear systems by elimination, including multiplying equations and recognizing special cases.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

Write the equations in aligned form, make one pair of coefficients opposites, and add the equations. One variable disappears, leaving a one-variable equation to solve.

01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Align like terms

Put x-terms, y-terms, and constants in consistent columns before combining equations.

2

Create opposites

Multiply every term of one or both equations so one variable has opposite coefficients.

3

Add, then back-substitute

Adding cancels the chosen variable. Solve what remains and use an original equation to find the other coordinate.

02

See the structure

A picture makes the relationships easier to remember.

The two lines intersect exactly at the point 6 comma 3.123456789-22468xyy = 9 − xy = x − 3(6, 3)
The graph uses a consistent coordinate scale: y = 9 − x and y = x − 3 intersect exactly at (6, 3).
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Ready to add

x+y=9\displaystyle x+y=9, x−y=3\displaystyle x-y=3

The two lines intersect exactly at the point 6 comma 3.123456789-22468xyy = 9 − xy = x − 3(6, 3)
Both graphs pass through (6, 3), confirming the elimination solution.
  1. Add the equations: 2x=12\displaystyle 2x=12.
  2. So x=6\displaystyle x=6.
  3. Then 6+y=9\displaystyle 6+y=9, so y=3\displaystyle y=3.
Answer(6,3)\displaystyle (6,3)
Example 2Multiply once

2x+3y=12\displaystyle 2x+3y=12, x+y=5\displaystyle x+y=5

  1. Multiply the second equation by −2: −2x−2y=−10\displaystyle -2x-2y=-10.
  2. Add to get y=2\displaystyle y=2.
  3. Then x+2=5\displaystyle x+2=5, so x=3\displaystyle x=3.
Answer(3,2)\displaystyle (3,2)
Example 3Multiply both

2x+3y=7\displaystyle 2x+3y=7, 3x+2y=8\displaystyle 3x+2y=8

  1. Multiply the first by 3 and second by −2.
  2. Add 6x+9y=21\displaystyle 6x+9y=21 and −6x−4y=−16\displaystyle -6x-4y=-16 to get 5y=5\displaystyle 5y=5.
  3. Then y=1\displaystyle y=1 and x=2\displaystyle x=2.
Answer(2,1)\displaystyle (2,1)
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Multiplying only one term instead of the entire equation.

  • Adding constants but subtracting variable terms inconsistently.

  • Forgetting to find the second variable after elimination.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Systems by Elimination

Practice cancellation with matched and unmatched coefficients.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Solve x+y=10\displaystyle x+y=10 and x−y=2\displaystyle x-y=2.

Show hint

Add the equations.

Show worked solution
  1. 2x=12\displaystyle 2x=12, so x=6\displaystyle x=6.
  2. Use 6+y=10\displaystyle 6+y=10.
  3. y=4\displaystyle y=4.

Answer: (6,4)\displaystyle (6,4)

2

Solve 2x+3y=12\displaystyle 2x+3y=12 and 2x−y=4\displaystyle 2x-y=4.

Show hint

Subtract the second equation.

Show worked solution
  1. Subtracting gives 4y=8\displaystyle 4y=8.
  2. y=2\displaystyle y=2.
  3. 2x−2=4\displaystyle 2x-2=4, so x=3\displaystyle x=3.

Answer: (3,2)\displaystyle (3,2)

3

Solve 3x+2y=16\displaystyle 3x+2y=16 and x+y=6\displaystyle x+y=6.

Show hint

Multiply the second equation by −2.

Show worked solution
  1. Use −2x−2y=−12\displaystyle -2x-2y=-12.
  2. Add to get x=4\displaystyle x=4.
  3. Then 4+y=6\displaystyle 4+y=6, so y=2\displaystyle y=2.

Answer: (4,2)\displaystyle (4,2)

4

Solve 2x+3y=13\displaystyle 2x+3y=13 and 3x−2y=0\displaystyle 3x-2y=0.

Show hint

Create opposite y-coefficients.

Show worked solution
  1. Multiply the first equation by 2 and the second by 3.
  2. Add 4x+6y=26\displaystyle 4x+6y=26 and 9x−6y=0\displaystyle 9x-6y=0: 13x=26\displaystyle 13x=26, so x=2\displaystyle x=2.
  3. Then 6−2y=0\displaystyle 6-2y=0, so y=3\displaystyle y=3.

Answer: (2,3)\displaystyle (2,3)

5

Classify 2x+4y=8\displaystyle 2x+4y=8 and x+2y=5\displaystyle x+2y=5.

Show hint

Double the second equation.

Show worked solution
  1. The second becomes 2x+4y=10\displaystyle 2x+4y=10.
  2. The same left side cannot equal both 8 and 10.
  3. Elimination produces 0=−2\displaystyle 0=-2.

Answer: No solution.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

Write the equations in aligned form, make one pair of coefficients opposites, and add the equations. One variable disappears, leaving a one-variable equation to solve.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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