Slope of a straight line

Do you have to calculate the slope of a straight line? Use our calculator and you will obtain both the slope and the equation of the line passing through two points.

All you have to do is write the coordinates of each point (x₁, y₁) (x₂, y₂) and press the calculate button to obtain all the data of the line they form.

What is the slope of a straight line?

Slope is defined as the degree of inclination of an element. From the geometrical point of view, the slope measures the inclination of the line with respect to the abscissa axis or X-axis..

The slope of a straight line is represented by the letter m and can be of different types:

• Positive: the line is ascending and the angle that the line forms with the x-axis is acute. If the angle increases, so does the slope. In this case, the value of m is always positive.
• Negativein this case the line is descending and the angle formed by the line with the x-axis is obtuse. If the angle increases, the slope decreases. Finally, in case of negative slope, the values of m will be negative.
• Zero or zero slopeIn these cases where m = 0 we can see how the line is parallel to the x-axis and, therefore, has no slope. In short, it is completely horizontal.

How to calculate the slope if we are given the angle

If we are given the angle that the straight line makes with the X axis, we can calculate the slope by applying the following formula:

m = tan θ

We simply have to calculate the tangent of the angle. As we said before, if the angle is:

• Acute (less than 90º)The slope is positive and increases as its value increases.
• Obtuse: (more than 90º)The slope is negative and decreases as its value increases.

To prove it, let's calculate the slope of an angle of 30º and another of 100º to verify if the above is true:

m = tan 30º = 0.57

m = tan 100º = -5.67

Indeed, we confirm that depending on the type of angle formed by the straight line with the X axis, ehe value of the slope can be negative or positive.

Formula for calculating the slope passing through two points

If we want to calculate the slope of a line passing through two pointsIf we do not use the formula above, we have to apply the formula you have above these lines.

Simply calculate the quotient of the y-axis increment by the x-axis increment. To understand it better, let's look at a exercise solved in which we are asked to calculate the slope of a line passing through the points (3,4) and (1,2).

We apply the formula and we have that:

m = (2 - 4) / (1 - 3) = -2/-2 = 1

Obtain the slope given the director vector of the line

The director vector of a straight line is defined as the vector that lies on or is parallel to this straight line. This means that what we are really interested in is the direction of the director vector.

For example, the director vector (2, 1) means that for every two units on the x-axis, we increase one unit on the y-axis. Also, as it is parallel to or inside the straight line itself, the slope will be the same.

¿How to calculate the slope in this case? We simply have to calculate this quotient:

m = v₂ / v₁

In the case of the example, the slope would be:

m = 1 /2 = 2

To get the slope if we are given the equation of the line

When give us the equation of the straight line, we can calculate the slope in different ways. It all depends on how the equation of the line is, so let's see the most common cases and how the slope is obtained in each of them.

With the continuous equation of the line

For example, if we are given the point-slope equation or continuous equation of the line: 

The slope will be m = v₂ / v₁

With the point-slope equation of the line

In this case, the formula can be this type:

y - y₁ = m (x - x₁)

y = mx + n

In both cases, the slope is the value of m.

With the general equation of the line

If we are given the general equation of the line (Ax + By + C = 0)we can calculate the slope with the following equation:

m = - A/B

Calculating the slope of a line with Excel

Do you know that you can calculate the slope of a line in Excel? You only need to know at least two points on the line to be able to do it.

If you know them, open a new spreadsheet and write in each cell the coordinates of each point.

When you have placed the points on the line, write the following function in Excel:

=SLEEP(y_coordinates;x_coordinates)

where coordinates_y and coordinates_x are the intervals of the cells in which you have written the positions of the points.

For a better explanation, we are going to see a practical example in which we are going to calculate the slope of a line passing through the points (1,3) and (5,6). We write them in Excel and we will have something like this:

Now we apply the Excel function that we have seen above and it would look like this:

=SLOPE(B2:B3;A2:A3)

As you can see, in the interval B2:B3 we have written the y-coordinates of the points on the line and in the interval A2:A3 we have the coordinates of the x-axis.

If after our post with the theory to calculate the slope of a straight line you still have doubts, leave us a comment and we will help you as soon as possible.

Nacho

Ingeniero de Telecomunicaciones dedicado al mundo de Internet. En esta web te ayudo a hacer cálculos y conversiones sencillas que con el paso de los años se nos ha olvidado cómo se hace. Si tienes dudas, déjame un comentario y te ayudaré. Si quieres mejorar tu proyecto online o necesitas asesoramiento, write me a message.