Algebra · Step-by-step guide

Slope-Intercept Form

Understand y = mx + b, write linear equations, and graph a line from its slope and y-intercept.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

In y=mx+b\displaystyle y=mx+b, m is the slope and b is the y-intercept (0,b)\displaystyle (0,b). Plot the intercept, use the slope as rise over run to find another point, then draw the line.

y=mx+b\displaystyle y=mx+b
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

m controls direction and steepness

The sign tells whether the line rises or falls; the magnitude tells how rapidly y changes for each unit of x.

2

b anchors the graph

The y-intercept is the output when x = 0 and gives the first convenient point.

3

Write from data

Find m first, then substitute a known point into y = mx + b to solve for b.

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 1Read the form

Describe y=−2x+5\displaystyle y=-2x+5.

  1. The coefficient of x is −2.
  2. The constant is 5.
  3. The graph falls 2 units for each 1 unit right and crosses at (0,5)\displaystyle (0,5).
AnswerSlope −2; y-intercept 5
Example 2Write from a point

Slope 3 through (2,1)\displaystyle (2,1).

  1. Start with y=3x+b\displaystyle y=3x+b.
  2. Substitute the point: 1=3(2)+b\displaystyle 1=3(2)+b.
  3. Solve b=−5\displaystyle b=-5.
Answery=3x−5\displaystyle y=3x-5
Example 3Convert standard form

Rewrite 2x+4y=12\displaystyle 2x+4y=12.

  1. Subtract 2x: 4y=−2x+12\displaystyle 4y=-2x+12.
  2. Divide every term by 4.
  3. Identify m and b.
Answery=−12x+3\displaystyle y=-\frac12x+3
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Plotting b on the x-axis.

  • Using run over rise for m.

  • Dividing only some terms when isolating y.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Slope-Intercept Form

Practice reading, writing, and graphing y = mx + b.

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

Identify the slope and y-intercept of y=3x−7\displaystyle y=3x-7.

Show hint

Compare with y=mx+b\displaystyle y=mx+b.

Show worked solution
  1. m\displaystyle m is the x-coefficient, so m=3\displaystyle m=3.
  2. b=−7\displaystyle b=-7.
  3. The intercept is (0,−7)\displaystyle (0,-7).

Answer: Slope 3; y-intercept (0,−7)\displaystyle (0,-7).

2

Write a line with slope −2 and y-intercept 5.

Show hint

Substitute m and b into the form.

Show worked solution
  1. Start with y=mx+b\displaystyle y=mx+b.
  2. Use m=−2\displaystyle m=-2 and b=5\displaystyle b=5.
  3. This gives y=−2x+5\displaystyle y=-2x+5.

Answer: y=−2x+5\displaystyle y=-2x+5

3

Write the line with slope 4 through (2,3)\displaystyle (2,3).

Show hint

Solve for b.

Show worked solution
  1. Use y=4x+b\displaystyle y=4x+b.
  2. 3=4(2)+b\displaystyle 3=4(2)+b, so b=−5\displaystyle b=-5.
  3. Write the completed equation.

Answer: y=4x−5\displaystyle y=4x-5

4

Rewrite 3x+6y=12\displaystyle 3x+6y=12 in slope-intercept form.

Show hint

Isolate y.

Show worked solution
  1. 6y=−3x+12\displaystyle 6y=-3x+12.
  2. Divide every term by 6.
  3. y=−12x+2\displaystyle y=-\frac12x+2.

Answer: y=−12x+2\displaystyle y=-\frac12x+2

5

Find the line through (−1,2)\displaystyle (-1,2) and (3,10)\displaystyle (3,10).

Show hint

Find m, then b.

Show worked solution
  1. m=(10−2)/(3−(−1))=8/4=2\displaystyle m=(10-2)/(3-(-1))=8/4=2.
  2. Use 2=2(−1)+b\displaystyle 2=2(-1)+b, so b=4\displaystyle b=4.
  3. Therefore y=2x+4\displaystyle y=2x+4.

Answer: y=2x+4\displaystyle y=2x+4

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

In y=mx+b\displaystyle y=mx+b, m is the slope and b is the y-intercept (0,b)\displaystyle (0,b). Plot the intercept, use the slope as rise over run to find another point, then draw the line.

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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