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Drag two points and watch the slope, the line and every form change.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
The slope of a line is how much it climbs for each step to the right: the change in y divided by the change in x, between any two points on it. Every rule about slope follows from that one sentence - a positive slope climbs, a negative one falls, a horizontal line has slope zero, and a vertical line has no slope at all because its run is zero and nothing can be divided by zero.
Which is easy to say and hard to feel from a formula. So this page draws it: the run along and the rise up, as a staircase between your two points. Drag one of them and watch which number changes. Doubling the rise doubles the slope; doubling the run halves it; moving a point along the line changes neither, which is the fact that makes slope worth defining in the first place.
You also get everything else two points determine, exactly: the equation of the line in all three forms your homework might ask for, the distance between the points as an exact radical, the midpoint, the intercepts, and the slope of any perpendicular line. Nothing is rounded on the way - the distance between (0,0) and (2,3) is √13, not 3.6056.
Fill in the four boxes, or drag either dot on the plane. Arrow keys work too, one unit at a time, so the plane is usable without a mouse.
The slope is at the top as an exact fraction with its decimal beside it. Under it are slope-intercept, point-slope and standard form of the same line - whichever your question asks for.
The dashed lines between your points are the run and the rise. If the fraction ever looks wrong, count the squares: the picture and the arithmetic have to agree.
Distance, midpoint, both intercepts, the angle with the x-axis and the perpendicular slope are all worked out below - these are the other things a two-point question usually wants.
The formulas behind every number on this page, with the case each one is for.
| What | Formula | Note |
|---|---|---|
| Slope | m = (y₂ - y₁)/(x₂ - x₁) | Rise over run, in that order |
| Slope-intercept form | y = mx + b | b is where the line crosses the y-axis |
| Point-slope form | y - y₁ = m(x - x₁) | Fastest from a point and a slope |
| Standard form | Ax + By = C | A, B, C whole numbers, A positive |
| Horizontal line | m = 0 | y = c |
| Vertical line | m undefined | x = c |
| Parallel lines | same m | Never meet |
| Perpendicular lines | m₁ × m₂ = -1 | Negative reciprocal |
| Distance | √((x₂-x₁)² + (y₂-y₁)²) | Pythagoras on the staircase |
| Midpoint | ((x₁+x₂)/2, (y₁+y₂)/2) | The average of each coordinate |
The rise is 2 - 8 = -6 and the run is 5 - 3 = 2, so the slope is -6/2 = -3. The line falls three units for every step to the right. Watch the order: taking the differences the other way round gives 6 and -2, which is the same slope - what you must not do is take y one way and x the other.
Rise 4, run 3, so the slope is 4/3 and stays a fraction. 1.33 is a rounding of it, and on a question about slope the fraction is the answer. The line's equation comes out as y = 4x/3 + 2/3, which is why standard form (4x - 3y = -2) is often the tidier way to write it.
Both points have x = 2, so the run is zero and the slope is undefined - not zero. The line is x = 2, which cannot be written as y = mx + b at all. This is the case a formula alone will not tell you about, and the page says it in words.
The rise is zero, so the slope is 0/8 = 0. The line is y = 4. Compare it with the vertical case above: zero on top gives a slope of zero, zero underneath gives no slope, and the two are as different as it gets.
