Algebra · Step-by-step guide

How to Solve Systems of Equations

Learn how to solve systems of equations by graphing, substitution, and elimination, with step-by-step examples and checks.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

A solution to a system is an ordered pair that makes every equation true. Graphing finds the intersection visually, substitution replaces one variable with an equal expression, and elimination combines equations to cancel a variable.

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

Identify the structure

Use graphing when a visual estimate matters, substitution when a variable is isolated, and elimination when coefficients already match or are easy to make opposite.

2

Solve for both variables

After finding one coordinate, substitute it into either original equation. A system answer needs both x and y.

3

Check in both equations

Substituting the ordered pair into both originals catches sign and arithmetic errors and confirms the shared solution.

02

See the structure

A picture makes the relationships easier to remember.

The two lines intersect exactly at the point 2 comma 3.-112345-2246xyy = 2x − 1y = −x + 5(2, 3)
The graph uses a consistent coordinate scale: y = 2x − 1 and y = −x + 5 intersect exactly at (2, 3).
03

Worked examples

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

Example 1Graphing

Solve y=2x−1\displaystyle y=2x-1 and y=−x+5\displaystyle y=-x+5.

The two lines intersect exactly at the point 2 comma 3.-112345-2246xyy = 2x − 1y = −x + 5(2, 3)
Both graphs pass through (2, 3), so that ordered pair is the solution.
  1. Graph y=2x−1\displaystyle y=2x-1: begin at the y-intercept (0,−1)\displaystyle (0,-1), then use slope 2 to plot (1,1)\displaystyle (1,1) and (2,3)\displaystyle (2,3).
  2. Graph y=−x+5\displaystyle y=-x+5: begin at (0,5)\displaystyle (0,5), then use slope −1 to plot (1,4)\displaystyle (1,4) and (2,3)\displaystyle (2,3).
  3. The two lines intersect at (2,3)\displaystyle (2,3), so that ordered pair solves both equations.
Answer(2,3)\displaystyle (2,3)
Example 2Substitution

Solve y=x+4\displaystyle y=x+4 and 2x+y=10\displaystyle 2x+y=10.

  1. Replace y with x+4\displaystyle x+4: 2x+x+4=10\displaystyle 2x+x+4=10.
  2. Solve 3x=6\displaystyle 3x=6, so x=2\displaystyle x=2.
  3. Use y=x+4\displaystyle y=x+4 to get y=6\displaystyle y=6.
Answer(2,6)\displaystyle (2,6)
Example 3Elimination

Solve 2x+y=11\displaystyle 2x+y=11 and 3x−y=9\displaystyle 3x-y=9.

  1. Add the equations; y cancels: 5x=20\displaystyle 5x=20.
  2. So x=4\displaystyle x=4.
  3. Substitute: 2(4)+y=11\displaystyle 2(4)+y=11, giving y=3\displaystyle y=3.
Answer(4,3)\displaystyle (4,3)
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Stopping after finding only one variable.

  • Changing an equation without applying the operation to every term.

  • Checking the answer in only one of the two equations.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Systems of Equations

Practice choosing and applying all three solution methods.

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 by graphing: y=x+2\displaystyle y=x+2 and y=−2x+8\displaystyle y=-2x+8.

Show hint

Plot each y-intercept, then follow its slope.

Show worked solution
  1. The first line passes through (0,2)\displaystyle (0,2) and rises 1 for every 1 right.
  2. The second passes through (0,8)\displaystyle (0,8) and falls 2 for every 1 right.
  3. Both graphs meet at (2,4)\displaystyle (2,4).

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

2

Solve by substitution: y=4x−5\displaystyle y=4x-5 and 2x+y=13\displaystyle 2x+y=13.

Show hint

Replace y with 4x−5\displaystyle 4x-5.

Show worked solution
  1. 2x+(4x−5)=13\displaystyle 2x+(4x-5)=13
  2. 6x=18\displaystyle 6x=18, so x=3\displaystyle x=3.
  3. y=4(3)−5=7\displaystyle y=4(3)-5=7.

Answer: (3,7)\displaystyle (3,7)

3

Solve by elimination: 3x+y=11\displaystyle 3x+y=11 and x−y=1\displaystyle x-y=1.

Show hint

Add the equations.

Show worked solution
  1. Adding cancels y: 4x=12\displaystyle 4x=12.
  2. x=3\displaystyle x=3.
  3. Substitute into x−y=1\displaystyle x-y=1: 3−y=1\displaystyle 3-y=1, so y=2\displaystyle y=2.

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

4

Classify y=−3x+2\displaystyle y=-3x+2 and 6x+2y=4\displaystyle 6x+2y=4.

Show hint

Rewrite the second equation.

Show worked solution
  1. Divide the second equation by 2: 3x+y=2\displaystyle 3x+y=2.
  2. Solve for y: y=−3x+2\displaystyle y=-3x+2.
  3. Both equations name the same line.

Answer: Infinitely many solutions.

5

Adult tickets cost $8 and student tickets cost $5. Twelve tickets cost $81. Find each number.

Show hint

Use a+s=12\displaystyle a+s=12 and 8a+5s=81\displaystyle 8a+5s=81.

Show worked solution
  1. Multiply a+s=12\displaystyle a+s=12 by 5: 5a+5s=60\displaystyle 5a+5s=60.
  2. Subtract from the cost equation: 3a=21\displaystyle 3a=21, so a=7\displaystyle a=7.
  3. Then s=12−7=5\displaystyle s=12-7=5.

Answer: 7 adult tickets and 5 student tickets.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

A solution to a system is an ordered pair that makes every equation true. Graphing finds the intersection visually, substitution replaces one variable with an equal expression, and elimination combines equations to cancel a variable.

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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