Slope: Rise Over Run, and What the Number Tells You

Slope is the number that answers two questions at once: how steep is this line, and which way does it lean? A road sign that says “8% grade” is reporting slope. So is the pitch of a roof, the gradient of a wheelchair ramp, and the rate on a pay-per-mile invoice.

For a straight line the slope is the same everywhere along it, which is what makes it a single, useful number. Pick any two points on the line, measure how much you go up and how much you go across between them, and divide.

The Slope Formula

The rise is the vertical change between two points; the run is the horizontal change. Slope is rise divided by run.

The Four Kinds of Slope

The sign and size of describe the line completely:

Two lines with slopes and both rise, but the first is four times as steep. Size is steepness; sign is direction.

Zero versus undefined trips people up. A horizontal line has slope — a perfectly real number. A vertical line has undefined slope — there is no number for it, because the run is . Saying a vertical line has “zero slope” is a specific, common error.

Why the Order of the Points Does Not Matter

Take and . Computing one way:

Computing with the points swapped:

Swapping the points flips the sign of the top and the bottom, and two sign flips cancel. The only rule is consistency: whatever order you subtract the -values, subtract the -values the same way. Mixing the order — over — gives the wrong sign and is the single most common slope mistake.

Reading Slope From a Graph

1. Find two points where the line passes cleanly through grid intersections.
2. Count the rise: how many units up (positive) or down (negative) from the left point to the right point.
3. Count the run: how many units right (positive) — moving left to right keeps the run positive so the sign lives entirely in the rise.
4. Divide rise by run and simplify.

If the line goes up as you read left to right, the slope is positive; if it goes down, negative. A quick gut check before you compute anything.

Slope as a Rate of Change

Slope is not only a geometry idea. Whenever a graph plots one real quantity against another, the slope is a rate:

• distance vs time → slope is speed (e.g. km per hour)
• cost vs number of items → slope is price per item
• water in a tank vs time → slope is fill rate (litres per minute)
• earnings vs hours worked → slope is the hourly wage

The units come straight from the axes: slope is measured in -units per one -unit. A line on a cost graph with slope means per item, and the steeper the line, the more each item adds.

Worked Example A: Positive Slope

Find the slope through and .

Positive, so the line rises. For every step right, it climbs two.

Worked Example B: Negative Slope

Find the slope through and .

Negative, so the line falls — down two for every one across.

Worked Example C: Zero Slope

Find the slope through and .

The -values are equal, so there is no rise. The line is horizontal, .

Worked Example D: Undefined Slope

Find the slope through and .

The -values are equal, so the run is zero. The line is vertical, , and has no slope.

Worked Example E: A Fractional Slope

Find the slope through and .

Leave it as . It means the line rises for every across — a gentle upward slope, less than .

Worked Example F: Slope as Speed

A car’s distance from home is km after hour and km after hours. What is the slope of the distance-time graph, and what does it mean?

The slope is , in km per hour — the car’s average speed is km/h.

Common Mistakes to Avoid

• Inconsistent subtraction order. mixes the order and flips the sign. Subtract both the same way.
• Run over rise. Slope is rise over run — -change on top.
• Calling a vertical line “zero slope.” Vertical is undefined; horizontal is zero.
• Dropping a sign when subtracting a negative. , not . Negatives in the coordinates cause most slope errors.
• Simplifying a fraction to a “nicer” wrong value. is , not .
• Reading rise and run off a graph without the scale. If one square is units, a rise of “3 squares” is .
• Assuming slope changes along a straight line. It does not — any two points give the same . If yours do not, one point is misread.

Where Slope Shows Up Next

• Lines. Slope-intercept form and point-slope form are both built around .
• Parallel and perpendicular lines. Parallel lines share a slope; perpendicular slopes are negative reciprocals.
• Systems of equations. Two lines with the same slope never meet (no solution) or coincide (infinitely many).
• Calculus. The derivative is the slope of a curve at a single point — the same rise-over-run idea taken to a limit.
• Applications. Speed, unit price, interest rate, and rate of flow are all slopes of the right graph.

Practice Problems

Work each one before opening the answer.

Problem 1. Slope through and .

Problem 2. Slope through and .

Problem 3. Slope through and .

Problem 4. Slope through and .

Problem 5. Slope through and .

Problem 6. Compute the slope through and both ways to show the order does not matter.

Problem 7. A line rises units for every units it runs. What is its slope, and is it steeper or gentler than ?

Problem 8. A phone plan costs for GB and for GB of data. Treating cost as a line against data, find the slope and say what it means.

Problem 9. The points , , are said to be on one straight line. Check using slope.

Problem 10. A ramp rises m over a horizontal distance of m. What is its slope, and as a percentage grade?

Problem 11. A line has slope and passes through . Find another point on it.

Problem 12. Two points on a line are and . Find the slope.

Quick Reference

Slope is the in every form of a linear equation — see Lines — and it is computed from coordinate differences on The Cartesian Coordinate System. It also gives the fastest way to sketch when graphing equations. More Algebra lessons are available too.