For a line, slope is “rise over run”; read it as m in y=mx+b, compute (y₂−y₁)/(x₂−x₁), or use −A/B from Ax+By=C.
Slope tells you how steep a line is and which way it tilts. Once you know where to look, you can pull slope straight from an equation, a graph, a table of values, or two points. This guide shows the exact steps for every common format, plus quick checks so you can spot mistakes before they snowball.
How To Find A Slope Of An Equation — With Points, Graphs, Or Forms
Different formats pack slope in different places. Use the map below to jump to the method you need, then walk through the short steps. If you came here wondering how to find a slope of an equation in a hurry, start with the first table, then follow the worked examples.
Fast Map: Where Slope Lives In Each Form
| Equation / Data Form | When You See It | How To Get Slope m |
|---|---|---|
| Slope-intercept: y = mx + b | Line already solved for y | Coefficient of x is m |
| Standard: Ax + By = C | Integers for A, B, C | m = −A/B (if B ≠ 0) |
| Point-slope: y − y₁ = m(x − x₁) | You know one point and m | m is written directly |
| Two points: (x₁, y₁), (x₂, y₂) | Only points given | m = (y₂ − y₁)/(x₂ − x₁), x₂ ≠ x₁ |
| Horizontal line | y = c | m = 0 |
| Vertical line | x = c | Undefined (no slope value) |
| Table of values | Pairs of (x, y) | Pick any two rows and compute Δy/Δx |
| Graph of a line | Picture only | Count “rise” over “run” between two clear points |
Finding The Slope Of An Equation: Step-By-Step
Below are short, test-ready procedures with common pitfalls to dodge. Keep scratch work tidy and label points so your signs don’t flip.
1) Read Slope From Slope-Intercept Form
Form: y = mx + b. The coefficient of x is the slope. No extra algebra needed.
Example: y = −3x + 7 → slope m = −3. If decimals or fractions appear, keep them as-is: y = 0.25x − 5 gives m = 0.25, and y = (5/3)x + 2 gives m = 5/3.
Watch for: Hidden 1’s. In y = x − 4, slope is 1, not x.
2) Convert Standard Form To Slope
Form: Ax + By = C. Solve for y or apply the shortcut m = −A/B as long as B ≠ 0.
Example: 3x + 2y = 10. Then m = −3/2. If you prefer to see it, isolate y: 2y = −3x + 10 → y = (−3/2)x + 5.
Edge cases: If B = 0, the line is vertical (x = C/A), slope undefined. If A = 0, the line is horizontal (y = C/B), slope 0.
3) Use Two Points To Compute Slope
Formula: label points in any order, then compute m = (y₂ − y₁)/(x₂ − x₁) with x₂ ≠ x₁.
Example: points (−2, 5) and (4, −1). Numerator: −1 − 5 = −6. Denominator: 4 − (−2) = 6. So m = −6/6 = −1.
Sign tips: Keep the order consistent. If you start with (x₁, y₁) on top, use the matching x₁ on the bottom.
4) Read Slope From A Graph
Pick two points where the line crosses grid intersections. From the left point to the right point, count vertical change (rise) and horizontal change (run). Slope is rise/run. If the line goes down as it moves right, the slope is negative.
Example: Moving from (1, 2) to (4, 5) rises 3 and runs 3, so m = 1. Moving from (−3, 4) to (0, −2) drops 6 while running 3, so m = −2.
5) Use Point-Slope Form When One Point And Slope Are Known
Form: y − y₁ = m(x − x₁). The slope is already there. If a different form is required, expand and rearrange.
Example: With slope m = 3/4 through (2, −1), the model is y + 1 = (3/4)(x − 2). Slope remains 3/4.
6) Read Slope From A Table Of Values
Pick two rows with different x. Compute Δy/Δx. If the function is linear, that ratio stays constant across rows.
Example: rows (x, y) = (1, 7) and (5, −1) give m = (−1 − 7)/(5 − 1) = −8/4 = −2.
Worked Examples You Can Copy
Example A: Slope From Slope-Intercept
Problem: Find the slope of y = (5/2)x − 11.
Step: Read the coefficient of x. That’s m = 5/2.
Check: Positive slope means the line rises left-to-right. That matches the sign.
Example B: Slope From Standard Form
Problem: Find the slope of 7x − 3y = 12.
Step 1: Use m = −A/B. Here A = 7, B = −3. So m = −7/(−3) = 7/3.
Step 2 (optional): Solve for y to verify: −3y = −7x + 12 → y = (7/3)x − 4. The coefficient of x again shows m = 7/3.
Example C: Slope From Two Points
Problem: Find the slope through (−4, 3) and (2, 15).
Step: m = (15 − 3)/(2 − (−4)) = 12/6 = 2.
Check: Going right by 6 raises 12, so 2 units up per 1 unit right fits.
Example D: Slope When A Line Is Horizontal Or Vertical
Horizontal: y = −9 → m = 0.
Vertical: x = 4 → slope is undefined (division by zero in the two-point formula).
Common Mistakes And Quick Fixes
Sign Slips
If your numerator and denominator both pick the second point minus the first, your signs align. Mixing orders flips the sign and breaks the answer. Write a tiny “−” arrow beside each subtraction to stay consistent.
Zero In The Denominator
If x₂ = x₁, you’re on a vertical line and slope doesn’t exist. That’s not an error; it’s a property. If a numeric slope is required, the problem isn’t linear or needs a different model.
Reducing Fractions
Slope can be left as a reduced fraction to preserve rise/run meaning. 8/12 should be written as 2/3. Decimal forms are fine on calculators, but fractions stay exact.
Units And Rate Language
In word problems, add units to slope. If y is dollars and x is hours, m is dollars per hour. That matches the “per” in rise per run.
Attach Slope To Real-World Models
Linear models show up in pricing, speed, and conversions. If a service charges a base fee plus a rate, you’re looking at y = mx + b. Slope is the rate. If a recipe scales servings by a fixed amount per unit, slope records that change. When data sit on a straight line, slope captures “how fast” one quantity responds to the other.
You can learn more techniques and see animated examples in Khan Academy’s slope guide, and you’ll find a classroom-ready overview with checks and practice in the OpenStax Algebra lesson on slope.
Minimal Algebra You Need For Each Form
From Standard To Slope-Intercept
Start with Ax + By = C. Move the x term: By = −Ax + C. Divide by B: y = (−A/B)x + C/B. The coefficient of x is your slope, so m = −A/B.
From Two Points To An Equation With Slope
Compute m = (y₂ − y₁)/(x₂ − x₁). Then plug one point into point-slope: y − y₁ = m(x − x₁). If you need slope-intercept, distribute and solve for y.
From A Table To Slope-Intercept
Pick any two rows, get m. Use any one row as (x₁, y₁), then solve y − y₁ = m(x − x₁) for y to reveal the intercept.
Quick Reference Cases
These snippets get you from raw input to slope without fuss. Keep them handy during practice.
| Scenario | Input Data | Slope m |
|---|---|---|
| Rising line | Any two points with larger y as x grows | Positive number |
| Falling line | Any two points with smaller y as x grows | Negative number |
| Flat line | y = c | 0 |
| Vertical line | x = c | Undefined |
| Standard form | Ax + By = C with B ≠ 0 | −A/B |
| Graph count | Two clear lattice points | rise/run |
| Table rows | Two distinct x values | Δy/Δx |
Practice Prompts With Hints
Prompt 1: Read From Slope-Intercept
y = −0.8x + 13. Slope is the x coefficient: −0.8.
Prompt 2: Convert From Standard
4x + y = −6. Either use −A/B = −4/1 = −4 or rearrange to y = −4x − 6, which shows the same slope.
Prompt 3: Two-Point Formula
(3, −7) and (9, −1) → m = (−1 − (−7))/(9 − 3) = 6/6 = 1.
Prompt 4: Table To Slope
Rows (−1, 2), (1, −2) → m = (−2 − 2)/(1 − (−1)) = −4/2 = −2.
FAQ-Style Checks Without The Fluff
Is Slope The Same No Matter Which Two Points I Pick?
On a straight line, yes. That’s the point of linearity. If your answers change, the data might not be linear or a point was copied wrong.
Can Slope Be A Fraction Or Decimal?
Yes. Keep it as a reduced fraction when you want “rise over run” clarity. Use decimals when a calculator or context expects them.
What If The Problem Gives Rate Language?
Phrases like “dollars per hour” or “miles per minute” already describe slope. Convert the words into numbers and units, and you’ve got m.
One Last Pass: The Two Lines You’ll See Most
Slope-Intercept: Quick Read
When a line is written as y = mx + b, the “how steep” part is sitting right in front of x. That’s why teachers love this form. If you need the intercept too, it’s the constant b.
Standard Form: Fast Shortcut
When a line is written as Ax + By = C with B ≠ 0, slope pops out as −A/B without a single extra step. If B = 0, stop and call it vertical. No number can stand in for that slope.
Your Takeaway
Pick the method that matches the input. Read m directly in y = mx + b, use −A/B in standard form, count rise/run on a graph, or compute (y₂ − y₁)/(x₂ − x₁) for pairs of points. If a friend asks how to find a slope of an equation next week, you’ll have a crisp answer in under a minute.