Find the slope, rise, run, and linear equation from two coordinate points. Use the graph to check your answer and learn how the formula works.
This tool is built for the most common geometry task: finding the slope of a straight line from two known points on the coordinate plane. You enter each point, click once, and the calculator returns the gradient, rise, run, and line equation.
Add the x and y coordinates for your first point. You can use whole numbers, negatives, or decimals.
Add the second point. The calculator compares the vertical change and horizontal change between both coordinates.
The formula uses rise over run, or (y2 - y1) / (x2 - x1), to find the steepness and direction of the line.
Check the slope value, line equation, and graph to confirm whether the line rises, falls, stays flat, or is undefined.
Keep the order of the points consistent. If you subtract the first point from the second point in the numerator, do the same in the denominator. That keeps the ratio correct. The sign of the result matters too. A positive number shows an upward incline from left to right, while a negative value shows a downward incline.
If both points are the same, there is no unique line to measure. If the x-values match, you have a vertical line and the slope is undefined. In that case, the graph is still useful because it shows why dividing by zero is not allowed.
Suppose your points are (2, 5) and (9, 19).
This tells you the line goes up 2 units for every 1 unit moved to the right, which means the line has a fairly steep positive gradient.
Your answer is more than one number. Each output explains how the line behaves and how you can use the result in geometry, algebra, graphing, and real-world measurement.
The slope is the main answer. It measures steepness, direction, and rate of change. A larger absolute value means a steeper line. A positive slope rises from left to right, a negative slope falls, zero slope is flat, and an undefined slope is vertical.
Rise is the vertical difference between the y-values. Run is the horizontal difference between the x-values. These values help you see the ratio behind the answer, which is useful in classwork, graph sketching, and checking whether your subtraction order was correct.
The equation output converts your coordinate result into a linear equation. That makes it easier to graph the line again, compare it with another equation, or move to a later step such as finding the y-intercept, x-intercept, or point-slope form.
The graph gives you a visual check. If the points look flat, the answer should be zero. If the line stands straight up, the slope should be undefined. If the line climbs quickly, expect a strong positive incline. Visual feedback catches input mistakes fast.
In many problems, the sign matters as much as the value. If a sales graph has a positive slope, growth is moving upward. If a road has a negative slope, the path is descending. If a line on a coordinate plane has zero slope, nothing changes vertically as x changes. That is why slope is often described as both a geometry idea and a rate of change idea.
You can also convert slope into other formats depending on your subject. Builders often want percent grade, engineers may use angle of incline, and algebra students may need the linear equation. All of those start with the same basic slope value.
If you want to calculate slope manually, the process is short and consistent. You only need two points and careful subtraction.
In this formula, m is the slope, y2 - y1 is the rise, and x2 - x1 is the run. The numerator measures vertical change, while the denominator measures horizontal change. Because slope is a ratio, both changes must be measured over the same pair of points.
Here is the manual process in plain language. First, write the two points clearly so you do not mix up the x-values and y-values. Second, subtract the first y-value from the second y-value. Third, subtract the first x-value from the second x-value. Last, divide the rise by the run and simplify if possible.
For example, with points (-4, 1) and (6, 11), the rise is 11 - 1 = 10 and the run is 6 - (-4) = 10. That gives a slope of 10 / 10 = 1. A slope of 1 means the line goes up 1 unit for every 1 unit to the right. On a graph, that creates a steady diagonal line.
If rise and run have the same sign, the line rises from left to right. This is common in growth graphs and upward ramps.
If rise and run have opposite signs, the line falls from left to right. This often appears in cooling charts, price drops, or descending grades.
A zero rise creates a horizontal line with slope 0. A zero run creates a vertical line with an undefined slope because division by zero is impossible.
Slope shows up in schoolwork, design plans, construction, data charts, and travel routes. These examples show how to apply the same formula in different settings.
Your teacher gives the points (1, 3) and (5, 11). The rise is 8 and the run is 4, so the slope is 2. That tells you the line gains 2 units of y for every 1 unit of x. Tip: write both points in the same order before subtracting so the sign stays correct.
Suppose a ramp rises 2.5 feet over a horizontal run of 30 feet. The slope is 2.5 / 30 = 0.0833, which is about an 8.33% grade. This is a good example of converting a slope to percent grade for accessibility planning. Tip: many building projects describe steepness as a percentage instead of a decimal.
A roof that rises 6 inches for every 12 inches of horizontal run has slope 6 / 12 = 0.5. That is a moderate incline. Roofers often describe this as a 6-in-12 pitch, while math class would call it a slope of one-half. Tip: pitch language and slope language describe the same geometry from different angles.
If a drainage line drops 1.2 feet over 40 feet, the slope is -1.2 / 40 = -0.03, or -3%. The negative sign matters because it shows the direction of the drop. Tip: when the project is about water flow, confirm whether the sign should be positive or negative based on the direction you are measuring.
A graph shows sales increasing from 120 units in week 2 to 200 units in week 6. The slope is (200 - 120) / (6 - 2) = 80 / 4 = 20. That means sales increased by 20 units per week across that interval. Tip: when slope appears on a chart, think of it as rate of change, not just a slanted line.
Once you know the slope, you can move beyond the raw ratio and turn it into a more useful algebra or measurement format.
Many ranking pages for this topic go further than basic slope. They also connect the slope to line equations, intercepts, and angle measures. That matters because in real assignments, the slope is often just the first step. After you find it, you may need to write the equation of the line, solve for the y-intercept, compare lines, or convert the result into a percent grade.
If you know one point and the slope, you can write the point-slope form as y - y1 = m(x - x1). Using the earlier example with slope 2 through point (2, 5), the equation becomes y - 5 = 2(x - 2). If you expand and simplify, you get y = 2x + 1. In that form, the y-intercept is 1, which means the line crosses the y-axis at (0, 1).
You can also estimate the x-intercept by setting y to 0 and solving the equation. For y = 2x + 1, the x-intercept is -0.5. These intercepts help when you graph a line quickly or compare one linear equation with another.
In construction or surveying, you may want an angle of incline instead of a pure ratio. The connection is m = tan(theta). If the slope is 0.5, then the angle of incline is arctan(0.5), which is about 26.57 degrees. If the slope is 1, the angle is 45 degrees. This is useful when you switch from coordinate geometry to measurement language used in roads, ramps, hills, and machine setup.
Percent grade is another common conversion. Multiply the slope by 100. A slope of 0.125 becomes a 12.5% grade. That format is often easier to read in drainage plans, trail maps, and road signs. Even though the wording changes from slope to grade or incline, the underlying idea is still the same ratio of vertical change to horizontal change.
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These answers cover the most common questions people ask about finding slope, graphing lines, and using rise over run.
Subtract y1 from y2 to get the rise, subtract x1 from x2 to get the run, and then divide rise by run. In formula form, slope m = (y2 - y1) / (x2 - x1).
An undefined slope means the run is zero because the x-values are the same. That creates a vertical line, so you cannot divide by zero to get a numeric slope.
Rise is the vertical change between the points, which is y2 - y1. Run is the horizontal change, which is x2 - x1. The ratio rise over run is the slope.
Yes. Multiply the decimal slope by 100 to convert it to percent grade. For example, a slope of 0.08 equals an 8% grade.
First find the slope. Then use either point-slope form, y - y1 = m(x - x1), or slope-intercept form, y = mx + b, after solving for the y-intercept b.
A horizontal line has no vertical change, so the rise is zero. When rise is zero and the run is not zero, the slope equals 0.
Yes. A negative slope means the line falls from left to right. As x increases, y decreases.
For a straight line, slope and rate of change describe the same idea: how much y changes for each 1-unit change in x. In real-life graphs, that can mean speed, cost per item, or temperature change over time.
Yes. You can use slope to compare height change with horizontal distance for roofs, ramps, roads, drainage lines, and landscape grading. You may still need to convert the result to pitch, angle, or percent grade depending on the job.