Slope Calculator

Q: What does a negative slope mean?

The line goes down as x increases. A slope of -3/2 drops 3 units for every 2 units to the right.

Q: How do I get the equation of the line?

Compute the slope, then use b = y_1 - m x_1 to find the y-intercept and write y = mx + b.

Q: How do I find a perpendicular slope?

Take the negative reciprocal of the original slope. If m = -3/2, a perpendicular line has slope 2/3, and the product of the two slopes is -1.

About the Slope Calculator

The slope of a line measures how steeply it rises or falls: the change in $y$ divided by the change in $x$ between any two points. A positive slope rises to the right, a negative slope falls, zero is horizontal, and a vertical line has an undefined slope.

Once the slope $m$ is known, the y-intercept $b$ follows from either point, giving the line $y = mx + b$ .

What is the slope of a line?

Slope is a single number that describes the steepness and direction of a straight line. It is defined as rise over run: how far the line goes up or down for every unit it moves to the right. A slope of 2 means the line climbs 2 units for each step right, a slope of $-\frac32$ means it falls 3 units for every 2 to the right, and a slope of 0 means it is flat.

Because a straight line has the same steepness everywhere, any two points on it give the same slope. That makes the slope the key to the line's equation: once you know $m$ and one point, the whole line is determined. In everyday terms slope is a rate of change, such as kilometres per hour on a distance-time graph or dollars per item on a cost graph.

How to use the Slope Calculator

Enter the coordinates of the first point (x₁, y₁).
Enter the coordinates of the second point (x₂, y₂).
Press Find slope to see rise, run, the reduced slope and the line equation.

Formulas

Slope

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

Slope-intercept form

y = mx + b,\quad b = y_1 - m x_1

Point-slope form

y - y_1 = m(x - x_1)

Worked example: (2, 3) and (6, 11)

The worked example uses the points (2, 3) and (6, 11). The rise is $11 - 3 = 8$ and the run is $6 - 2 = 4$ , so the slope is $\frac84 = 2$ . Using the first point, the intercept is $b = 3 - 2 \cdot 2 = -1$ , giving the line $y = 2x - 1$ . The check substitutes the second point: $2 \cdot 6 - 1 = 11$ , as expected.

The solution below is generated by the same engine that powers the calculator above.

Final answer

m = 2,\quad y = 2x - 1

How we read your problem

Find the slope (rate of change) of the line through two points.

Step-by-step solution

Label the points
$(x_1, y_1) = (2, 3),\quad (x_2, y_2) = (6, 11)$
Find the change in y (rise)
$y_2 - y_1 = 11 - 3 = 8$
Find the change in x (run)
$x_2 - x_1 = 6 - 2 = 4$
Divide rise by run
$m = \frac{8}{4} = 2$
Find the y-intercept using the first point
$b = y_1 - m x_1 = 3 - 2 \cdot 2 = -1$
Write the equation of the line
$y = 2x - 1$

Formulas used

Slope
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Slope-intercept form
$y = mx + b$

The slope is 2: for every 1 unit increase in x, y changes by 2. The line rises from left to right.

Verification

Second point: 2·(6) + -1 = 11 (expected 11)

Detailed explanation

Negative, fractional and undefined slopes

For the points (1, 2) and (5, −4) the rise is $-6$ and the run is 4, so the calculator reports $m = -\frac32$ and the line $y = -\frac32 x + \frac72$ , keeping both values as exact fractions instead of −1.5 and 3.5. For (3, 1) and (3, 7) the run is zero; division by zero is not allowed, so the slope is undefined and the calculator reports the vertical line $x = 3$ instead of a $y = mx + b$ equation.

Three ways to write the line

Slope-intercept form $y = mx + b$ shows the slope and the y-intercept at a glance and is the form most graphing tools expect. Point-slope form $y - y_1 = m(x - x_1)$ is quicker to write when you have a point and a slope but have not found the intercept yet. Standard form $Ax + By = C$ with integer coefficients is preferred in some courses; from $y = 2x - 1$ it is $2x - y = 1$ . All three describe the same set of points.

Slope as a rate of change

In applied problems the slope carries units: the units of y divided by the units of x. If a taxi charges 11 dollars for a 6 km ride and 3 dollars for a 2 km ride, the slope through (2, 3) and (6, 11) is 2 dollars per kilometre, and the intercept of −1 is the value of the line at zero distance. Read that way, the slope answers the question how much does y change when x increases by one, which is the same idea that the derivative extends to curves.

Common mistakes

Subtracting in a different order on top and bottom, for example $y_2 - y_1$ over $x_1 - x_2$ .
Writing run over rise instead of rise over run.
Calling the slope of a vertical line 0; it is undefined.
Using the coordinates of one point as if they were rise and run.
Simplifying $\frac{8}{4}$ wrongly or dropping a minus sign from the run when the x-values decrease.

Tips

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.
Keep the slope as a fraction to avoid rounding when finding the intercept.
Plot both points on the graphing calculator to sanity-check the sign.
Before calculating, picture the points: if the second is higher and to the right, the slope must be positive.
The slope of a perpendicular line is the negative reciprocal: a line perpendicular to $y = 2x - 1$ has slope $-\frac12$ .

Frequently asked questions

How do I find the slope between two points?

Subtract the y-values, subtract the x-values in the same order, and divide. For (2, 3) and (6, 11): $m = \frac{11-3}{6-2} = 2$ .

What does a negative slope mean?

The line goes down as x increases. A slope of $-\frac32$ drops 3 units for every 2 units to the right.

Why is the slope of a vertical line undefined?

Both points have the same x, so the run is 0 and the formula would divide by zero. The line is written as $x = c$ instead.

How do I get the equation of the line?

Compute the slope, then use $b = y_1 - m x_1$ to find the y-intercept and write $y = mx + b$ .

Does the order of the points matter?

No. Swapping the points changes the sign of both differences, so the quotient is the same.

Can I enter decimal coordinates?

Yes. Decimals are converted to exact fractions so the slope and intercept stay exact.

What is the slope of a horizontal line?

Zero. Both points have the same y-value, so the rise is 0 and $\frac{0}{\text{run}} = 0$ . The line is written $y = c$ .

How do I find a perpendicular slope?

Take the negative reciprocal of the original slope. If $m = -\frac32$ , a perpendicular line has slope $\frac23$ , and the product of the two slopes is $-1$ .

How can I tell if two lines are parallel?

Compare their slopes. Non-vertical lines are parallel exactly when their slopes are equal and their intercepts differ; $y = 2x - 1$ and $y = 2x + 4$ never meet.

Find the slope of your line

Enter two points above to get the slope as an exact fraction, the y-intercept and the equation of the line through them.

Use the Slope Calculator Ask the AI Math Solver