Algebra · Step-by-step guide

How to Find Slope

Find slope from two points, a graph, a table, or an equation and understand what the result means.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

Slope is vertical change divided by horizontal change: rise over run. From two points, use m=(y2−y1)/(x2−x1)\displaystyle m=(y_2-y_1)/(x_2-x_1) and keep the subtraction order consistent.

m=y2−y1x2−x1\displaystyle m=\frac{y_2-y_1}{x_2-x_1}
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

Measure change

The numerator records how much y changes; the denominator records how much x changes.

2

Read the sign

Positive slope rises left to right, negative slope falls, zero slope is horizontal, and a vertical line has undefined slope.

3

Connect representations

In a table, compare changes; on a graph, count rise and run; in y = mx + b, read the coefficient of x.

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 1Two points

Find the slope through (2,3)\displaystyle (2,3) and (6,11)\displaystyle (6,11).

  1. Compute the rise: 11−3=8\displaystyle 11-3=8.
  2. Compute the run: 6−2=4\displaystyle 6-2=4.
  3. Divide: m=8/4\displaystyle m=8/4.
Answerm=2\displaystyle m=2
Example 2From an equation

Find the slope of 3x+2y=8\displaystyle 3x+2y=8.

  1. Solve for y: 2y=−3x+8\displaystyle 2y=-3x+8.
  2. Divide by 2: y=−32x+4\displaystyle y=-\frac32x+4.
  3. The x-coefficient is the slope.
Answerm=−32\displaystyle m=-\frac32
Example 3From a table

x increases 2 while y decreases 6.

  1. Use change in y over change in x.
  2. m=−6/2\displaystyle m=-6/2.
  3. A negative result matches the decreasing outputs.
Answerm=−3\displaystyle m=-3
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Reversing the subtraction order in only the numerator.

  • Using x-change over y-change.

  • Calling a vertical line’s slope zero instead of undefined.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Slope Practice

Find and interpret slope from graphs, points, tables, and equations.

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

Find the slope through (1,2)\displaystyle (1,2) and (5,10)\displaystyle (5,10).

Show hint

Use change in y over change in x.

Show worked solution
  1. Rise: 10−2=8\displaystyle 10-2=8.
  2. Run: 5−1=4\displaystyle 5-1=4.
  3. m=8/4=2\displaystyle m=8/4=2.

Answer: m=2\displaystyle m=2

2

Find the slope through (−3,4)\displaystyle (-3,4) and (2,−6)\displaystyle (2,-6).

Show hint

Keep the subtraction order consistent.

Show worked solution
  1. Deltay=−6−4=−10\displaystyle Delta y=-6-4=-10.
  2. Deltax=2−(−3)=5\displaystyle Delta x=2-(-3)=5.
  3. m=−10/5=−2\displaystyle m=-10/5=-2.

Answer: m=−2\displaystyle m=-2

3

Find the slope of 4x−2y=8\displaystyle 4x-2y=8.

Show hint

Solve for y.

Show worked solution
  1. −2y=−4x+8\displaystyle -2y=-4x+8.
  2. Divide by −2: y=2x−4\displaystyle y=2x-4.
  3. The x-coefficient is 2.

Answer: m=2\displaystyle m=2

4

Find the slope of the horizontal line y=7\displaystyle y=7.

Show hint

Compare any two points on the line.

Show worked solution
  1. Use (0,7)\displaystyle (0,7) and (3,7)\displaystyle (3,7).
  2. The rise is 7−7=0\displaystyle 7-7=0.
  3. m=0/3=0\displaystyle m=0/3=0.

Answer: m=0\displaystyle m=0

5

Find the slope of the vertical line x=−4\displaystyle x=-4.

Show hint

Notice that the run is zero.

Show worked solution
  1. Use (−4,0)\displaystyle (-4,0) and (−4,5)\displaystyle (-4,5).
  2. The run is −4−(−4)=0\displaystyle -4-(-4)=0.
  3. Division by zero is undefined.

Answer: Undefined slope.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

Slope is vertical change divided by horizontal change: rise over run. From two points, use m=(y2−y1)/(x2−x1)\displaystyle m=(y_2-y_1)/(x_2-x_1) and keep the subtraction order consistent.

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.

Keep learning

Related guides