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# Slope of a Line - Formula, Calculation, Examples

**TL;DR**

The slope of a line - sometimes called the gradient — measures the line's steepness as the ratio of vertical change to horizontal change between any two points: m=\frac{y_2 - y_1}{x_2 - x_1}, or "rise over run."

**BT**

**Last updated on June 9, 2022** 9 min read

## What Is the Slope of a Line?

The **slope** of a line is a number that describes its steepness and direction. Geometrically, it's the ratio of _rise_ (vertical change) to _run_ (horizontal change) between any two points on the line.

The formula, given two points (x1,y1) and (x2,y2):

m=\frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}

The Greek letter Δ (delta) is shorthand for _"change in._" The slope is the change in y per unit change in x — what mathematicians and physicists call the _rate of change_.

## What Is the Slope Formula?

Given any two points (x1,y1) and (x2,y2) on a line:

m=\frac{y_2 - y_1}{x_2 - x_1}

**Worked example.** Find the slope of the line through (2,3) and (6,7).

m=\frac{7 - 3}{6 - 2} = \frac{4}{4} = 1

The slope is **1** — the line rises 1 unit for every 1 unit right.

**Worked example.** Find the slope through (2,3) and (5,11).

m=\frac{11 - 3}{5 - 2} = \frac{8}{3}

The slope is **\frac{8}{3}** — the line rises 8 units for every 3 units right (a steep climb).

## What Are the Four Types of Slope?

The sign and form of the slope tell you the line's direction at a glance.

| Type        | Slope Value | What the Line Does                    |
|-------------|-------------|---------------------------------------|
| **Positive**| m > 0      | Rises from left to right              |
| **Negative**| m < 0      | Falls from left to right              |
| **Zero**    | m = 0      | Horizontal line (y = constant)        |
| **Undefined**| Δx = 0    | Vertical line (x = constant)         |

### Positive Slope

Line goes _up_ as you move right. Example: m=2, m=\frac{1}{3}.

### Negative Slope

Line goes _down_ as you move right. Example: m=-3, m=-\frac{2}{5}.

### Zero Slope (Horizontal Line)

Horizontal line. y=5 has slope 0 because there's no vertical change as x varies.

### Undefined Slope (Vertical Line)

Vertical line. x=3 has _undefined_ slope because the denominator x2−x1=0, and division by zero is undefined.

## How Do You Find Slope From a Graph?

Pick any two clearly-marked points on the line. Count vertical units between them (the rise) and horizontal units (the run). Slope is rise divided by run.

**Tip.** Choose points with integer coordinates whenever possible — they make the arithmetic cleaner.

**Tip.** The slope is the same between _any_ two points on the line. Pick the easiest two.

## What Is the Slope of Parallel Lines?

Two non-vertical lines are **parallel** if and only if they have **equal slopes**.

m1=m2

The reason is geometric: parallel lines have the same direction — they never meet — so the rise-per-run ratio is identical for both.

**Worked example.** The line y=3x+4 is parallel to y=3x−7. Both have slope 3. They tilt up at the same rate; only the y-intercept differs.

**Worked example.** Is the line through (1,2) and (4,11) parallel to y=3x+1?

Slope through the two points: m=\frac{11 - 2}{4 - 1} = \frac{9}{3} = 3.

Slope of y=3x+1: m=3.

Equal slopes ✓ — the lines are **parallel**.

**Edge case.** Two vertical lines (each with undefined slope) are parallel to each other — but the rule "equal slopes" can't be checked because the slope value doesn't exist. Treat parallel vertical lines as a separate case.

## What Is the Slope of Perpendicular Lines?

Two non-vertical lines are **perpendicular** if and only if the **product of their slopes is −1** — that is, their slopes are **negative reciprocals** of each other.

m1⋅m2=−1

Geometrically, rotating a line 90° flips its rise-and-run _and_ inverts the ratio with a sign change — that's where the negative reciprocal comes from.

**Worked example.** A line has slope 2. The slope of any line perpendicular to it is −\frac{1}{2}. Check: 2×−\frac{1}{2}=−1 ✓.

**Worked example.** A line has slope −\frac{3}{4}. The slope of any line perpendicular to it is \frac{4}{3}. Check: −\frac{3}{4}×\frac{4}{3}=−1 ✓.

**Worked example.** Is the line y=\frac{1}{2}x+3 perpendicular to y=−2x+5?

m1=\frac{1}{2}, m2=−2. Product: \frac{1}{2}×(−2)=−1 ✓ — yes, **perpendicular**.

## Why Does Slope Matter? (The Real-World GROUND)

The slope concept appears everywhere a _rate_ is measured. Some examples:

- **Road grade.** A road labeled _"7% grade"_ has a slope of 0.07 — it rises 7 metres per 100 metres horizontal.

- **Wheelchair ramp standards.** The Americans with Disabilities Act (ADA) specifies a maximum ramp slope of 1:12 (about m≈0.083). Steeper ramps are unsafe.

- **Speed and velocity.** On a position-vs-time graph, the slope _is_ the velocity. A horizontal line means stationary; a steep upward line means fast motion.

- **Linear regression.** In statistics, the slope of the best-fit line tells you how much y changes per unit increase in x — the _regression coefficient_.

- **Skiing and roof pitch.** A "double black diamond" ski run typically has a slope of tan(35°)≈0.7 or steeper.

The concept comes from René Descartes and Pierre de Fermat, who independently invented **analytic geometry** in the 1630s.

## A Worked Example

Find the slope of the line through (−2,5) and (4,−7).

**The intuitive (wrong) approach.** A student mixes up the subtraction order:

m=?−7−5−2−4=−12−6=2

The answer is wrong.

**The correct method.**

m=\frac{y_2 - y_1}{x_2 - x_1} = \frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2

The slope is **−2** — the line falls 2 units for every 1 unit right.

## What Are the Most Common Mistakes With Slope?

### **Mistake 1: Flipping numerator and denominator order inconsistently**

**Where it slips in:** Computing slope from two points with the wrong subtraction order.

**The correct way:** Pick one order — \frac{y_2 - y_1}{x_2 - x_1} — and stick with it on both.

### **Mistake 2: Calling a vertical line "zero slope"**

**Where it slips in:** Confusing vertical (undefined) with horizontal (zero).

### **Mistake 3: Dividing rise/run as a decimal when fraction is exact**

**Where it slips in:** Reporting m≈0.667 instead of m=\frac{2}{3}.

## The Mathematicians Who Shaped Slope

**René Descartes (1596–1650)** — Invented analytic geometry in 1637. The concept of slope as a number depends entirely on his coordinate system.

**Pierre de Fermat (1607–1665)** — Independently developed analytic geometry around the same time as Descartes.

**Isaac Newton (1643–1727)** — Generalised the slope concept to curves via differential calculus.

## A Practical Next Step

1. Find the slope through (1,4) and (5,12).
2. Find the slope through (−3,7) and (2,−3).
3. Find the slope of y=6. Then find the slope of x=−2.

If problem 2 had a sign issue, return to the wrong-path-first example — consistent subtraction order is the trap.