Slope Formula: How to Find Slope From Two Points, a Graph or a Table

Why the slope of a line is rise over run

Slope tells you how steep a line is. Take one step to the right along the line. How far does it go up or down? That’s the slope.

Picture a phone plan that costs $45 a month plus $6 for each extra gig. Every extra gig is one step to the right. Every step adds $6, so the line climbs 6. The slope is 6.

That’s why any two points on a line give the same slope. The rise is how far you go up or down, which is y₂ − y₁. The run is how far you go across, which is x₂ − x₁. Divide the rise by the run.

Steepness works the same way. A ramp that rises 2 feet over a 24-foot run has a slope of 2 ÷ 24, or 1/12. Work it backward: a ramp with that slope that has to climb 3 feet needs a 36-foot run.

Bottom line: slope is how far the line climbs for each step across.

How to find slope from two points

This is the formula on your sheet. Here’s how it works:

1. Call one point (x₁, y₁) and the other (x₂, y₂). Either one can go first.
2. Subtract the y-values: y₂ − y₁. That’s the rise.
3. Subtract the x-values in the same order: x₂ − x₁. That’s the run.
4. Divide the rise by the run, then simplify.

The slope is the price of one more gig. When a question asks for a rate, like dollars per gig or miles per hour, it’s asking for a slope.

Slope from a graph: count the squares

On a graph, you can count instead of using the formula. Find two points where the line crosses a grid corner exactly. Start at the one on the left. Count up or down to the other point’s height, then count across to it.

Counting up gives a positive rise, and counting down gives a negative one. The run is positive, since you move right.

That last check takes two seconds. It rules out 3/2, −2/3 and −3/2 before you pick an answer.

Slope from a table: watch how x jumps

A table is a list of points, so pick any two rows and use the formula. The catch? The x-values don’t always go up by 1.

If you read −4 off the table as the slope, you’d say the tank drops 4 gallons an hour. That’s double the real rate, so always divide by how far x jumps.

Positive, negative, zero and undefined slope

• Positive: the line goes uphill, like a bill that grows with every gig.
• Negative: it goes downhill, like gas draining from a tank.
• Zero: it’s flat, like an unlimited plan that costs the same however much you use.
• Undefined: it goes straight up and down, like a wall. There’s no step across at all.

4 mistakes that cost points

Mistake 1: subtracting in a different order

Whichever point goes first on top goes first on the bottom too. For (1, 2) and (4, 8), the slope is (8 − 2) ÷ (4 − 1) = 2. Mix the order, (8 − 2) ÷ (1 − 4), and you get −2. Now the line goes the wrong way.

Mistake 2: putting the run on top

The y-values go on top. A line through (0, 0) and (2, 6) has a slope of 6 ÷ 2 = 3. Flip it and you get 1/3, a line that’s far too flat.

Mistake 3: losing a minus sign

Put negative numbers in parentheses. For (−2, 5) and (4, −7), the run is 4 − (−2) = 6, because subtracting a negative adds. So the slope is −12 ÷ 6 = −2. Drop that sign and you get a run of 2 and a slope of −6. On the TI-30XS, type −2 with the (−) key.

Mistake 4: mixing up zero and undefined

Through (1, 4) and (6, 4), the rise is 0, so the slope is 0 ÷ 5 = 0. Through (3, 1) and (3, 5), the run is 3 − 3 = 0. You can’t divide by 0, so that slope is undefined. Flat is zero. Straight up is undefined.

Practice: 5 new questions

Every set is new. Check each answer as you go, and read the working on any you miss before the next one.

Want more? Practice all of functions and graphing (25 questions, every skill in the topic), or take the full GED math practice test.