Algebra · Step-by-step guide

How to Graph Linear Equations

Graph linear equations using slope-intercept form, intercepts, and tables, with examples for each method.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

A line needs two points. You can plot the y-intercept and use slope, find both axis intercepts, or make a table of input-output pairs. Draw one straight line through the points.

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

Slope and intercept

For y = mx + b, begin at (0,b) and repeat the rise/run pattern.

2

Intercept method

Set y = 0 for the x-intercept and x = 0 for the y-intercept. This is especially efficient in standard form.

3

Table method

Choose convenient x-values, calculate y, and plot the resulting ordered pairs.

02

See the structure

A picture makes the relationships easier to remember.

run = 3rise = 2
Both marked points sit on grid intersections. Moving 3 units right and 2 units up gives slope 2/3.
03

Worked examples

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

Example 1Slope-intercept

Graph y=23x−1\displaystyle y=\frac23x-1.

  1. Plot (0,−1)\displaystyle (0,-1).
  2. Use rise 2 and run 3 to reach (3,1)\displaystyle (3,1).
  3. Draw the line through both points.
AnswerLine through (0,−1)\displaystyle (0,-1) and (3,1)\displaystyle (3,1)
Example 2Intercepts

Graph 2x+3y=6\displaystyle 2x+3y=6.

  1. Set y = 0: 2x=6\displaystyle 2x=6, so (3,0)\displaystyle (3,0).
  2. Set x = 0: 3y=6\displaystyle 3y=6, so (0,2)\displaystyle (0,2).
  3. Connect the intercepts.
AnswerLine through (3,0)\displaystyle (3,0) and (0,2)\displaystyle (0,2)
Example 3Vertical line

Graph x=−4\displaystyle x=-4.

  1. Every point has x-coordinate −4.
  2. Plot (−4,0)\displaystyle (-4,0) and (−4,2)\displaystyle (-4,2).
  3. Draw the vertical line.
AnswerVertical line at x=−4\displaystyle x=-4
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Drawing a curved path through the points.

  • Reading a negative slope as positive.

  • Trying to write a vertical line in y = mx + b form.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Graphing Lines

Practice graphing from several equation forms.

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

Graph y=2x−3\displaystyle y=2x-3. Name two points.

Show hint

Start at the y-intercept.

Show worked solution
  1. Plot (0,−3)\displaystyle (0,-3).
  2. Use slope 2/1\displaystyle 2/1: move right 1 and up 2.
  3. The next point is (1,−1)\displaystyle (1,-1).

Answer: Line through (0,−3)\displaystyle (0,-3) and (1,−1)\displaystyle (1,-1).

2

Graph y=−12x+4\displaystyle y=-\frac12x+4. Name two points.

Show hint

Use run 2 and rise −1.

Show worked solution
  1. Plot (0,4)\displaystyle (0,4).
  2. Move right 2 and down 1.
  3. Plot (2,3)\displaystyle (2,3) and draw the line.

Answer: Line through (0,4)\displaystyle (0,4) and (2,3)\displaystyle (2,3).

3

Graph 2x+5y=10\displaystyle 2x+5y=10 using intercepts.

Show hint

Set one variable to zero at a time.

Show worked solution
  1. Set y=0\displaystyle y=0: 2x=10\displaystyle 2x=10, so (5,0)\displaystyle (5,0).
  2. Set x=0\displaystyle x=0: 5y=10\displaystyle 5y=10, so (0,2)\displaystyle (0,2).
  3. Connect the two intercepts.

Answer: Line through (5,0)\displaystyle (5,0) and (0,2)\displaystyle (0,2).

4

Graph x=3\displaystyle x=3.

Show hint

All points share the same x-coordinate.

Show worked solution
  1. Plot (3,0)\displaystyle (3,0) and (3,2)\displaystyle (3,2).
  2. Draw a vertical line through them.
  3. Its slope is undefined.

Answer: Vertical line at x=3\displaystyle x=3.

5

Make a table for y=x−2\displaystyle y=x-2 using x=−1,0,2\displaystyle x=-1,0,2.

Show hint

Substitute each input.

Show worked solution
  1. At x=−1\displaystyle x=-1, y=−3\displaystyle y=-3.
  2. At x=0\displaystyle x=0, y=−2\displaystyle y=-2.
  3. At x=2\displaystyle x=2, y=0\displaystyle y=0.

Answer: Points (−1,−3)\displaystyle (-1,-3), (0,−2)\displaystyle (0,-2), and (2,0)\displaystyle (2,0).

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

A line needs two points. You can plot the y-intercept and use slope, find both axis intercepts, or make a table of input-output pairs. Draw one straight line through the points.

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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