Slope of a line formula and equations, representation, examples

The slope of a line is the tangent of the angle θ that said line forms with the horizontal axis, which by convention is measured counterclockwise. The slope of any line is always constant and that is why it is one of its most essential characteristics..

To calculate it, you need to know two points of the line, whose coordinates are (x₁,Y₁) and (x_two,Y_two). A segment that belongs to the line is drawn between both points and then the segments that represent the distance between x are drawn.₁ and x_two, and between and₁ and and_two, as in the figure below.

The three segments make up a right triangle whose legs are: Δx = x_two - x₁  y Δy = y_two - Y₁. They correspond respectively to a horizontal and a vertical displacement..

Now we define a quotient, called the tangent of the angle θ and abbreviated tg θ, which is precisely the slope m of the straight:

m = tg θ = Δy / Δx

Note that for a line, this angle remains constant, regardless of the points taken to calculate its tangent. In any case, this value offers us a measure of how steep the line is.

Through the coordinates of the selected points, the formula for the slope is:

m = (y - y₁ ) / (x_two - x₁)

Article index

• 1 Graphic representation
• 2 Types of slope
• 3 How do you calculate the slope of a line?
• 4 Solved exercises
  • 4.1 - Exercise 1
  • 4.2 - Exercise 2
• 5 Examples
  • 5.1 Example 1
  • 5.2 Example 2
• 6 References

Graphic representation

Below we have several situations in which the concept of slope is relevant. Its value can be easily calculated by measuring the respective vertical and horizontal displacement, and then making the quotient indicated at the beginning.

This gives us an idea of ​​the unevenness or decline of some structure, such as a ramp, a roof or a road:

The slope of the ramp shown in figure 2 on the left is m = 1/12, that of the roof is m = 1/3 and that of the road is expressed as a percentage. A percentage of 10% means that for every 100 meters that advance horizontally, 10 meters in height are gained:

In this case, the slope is 10/100 = 0.1, which, expressed as a percentage, equals 10%..

Slope types

The slope of a line can be positive, negative, or zero. For example, the line shown in figure 1 has a positive slope. We appreciate it immediately because we see that the line "rises" if we see it from left to right.

If the line descends from left to right, then its slope is negative. And when a line is horizontal, its slope is zero.

Finally, for vertical lines, the slope is not defined.

The graphic representation of each type is below:

How do you calculate the slope of a line?

Calculating the slope is very simple, you just have to find the vertical displacement and the horizontal displacement, and then make the quotient between both.

When we have the drawing of the line in the Cartesian plane, these displacements are found by choosing any two points on the line P₁ And p_two, determining its coordinates and applying the definition given at the beginning:

m = (y - y₁ ) / (x_two - x₁ )

Since the value of the slope is independent of the choice of P₁ And p_two , we are going to choose a point P any of coordinates (x, y) that belongs to the line, whose coordinates are not known, and another point P₁ whose coordinates are: (x₁,Y₁).

The slope is:

m = (y - y₁) / (x - x₁)

We can clear the Y:

and - and₁ = m (x - x₁)

Now suppose that point P₁ is the intersection of the line with the vertical axis, of coordinates (0, b). Substituting this in the above equation:

y - b = m (x - 0) → y = mx + b

This expression is known as the equation of the line in the form slope - intercept, since the line is unequivocally determined when its slope and its intersection with the vertical axis are known.

Knowing only the slope is not enough to characterize a line on the plane, since infinite lines could have the same slope, which means that they are parallel, but pass through other points.

Solved exercises

- Exercise 1

Find the slope of the line shown in the following figure:

Solution

P₁ And p_two are two easy-to-read points that will be used for the calculation, note also that they are the respective intersections with the coordinate axes.

The coordinates of each point are:

P₁ (4.0) and P_two (0.4)

By substituting into the equation for the slope:

m = (4 - 0) / (0 - 4) = 4 / (- 4) = -1

The slope is negative, which was to be expected after looking at the graph.

- Exercise 2

Find the equation of the line that passes through the point (1, -6) and is parallel to the line y = 2x - 3.

Solution

The slope of the line sought must be the same as that of y = 2x - 3, since they are parallel. For this line the slope is m = 2, therefore the one we are looking for has the form:

and - and₁ = 2 (x - x₁)

Now we substitute the point through which our line passes: x₁ = 1 and y₁ = -6.

y - (-6) = 2 (x - 1)

Therefore y = 2x - 2 - 6 → y = 2x - 8

Examples

Two quantities can be related in such a way that their graph is a straight line. In this case it is said that the quantities have linear dependence and the slope of the line can be interpreted as the rate of change from one variable to the other.

Example 1

Suppose a pool is filled with water at a rate constant in time. Naturally, the more time passes, the more water is stored. Well, the rate at which the pool is filled is precisely the slope of the line that relates the volume to time:

In this example, the pool is filling at the rate of 6/3 gallons per minute or 2 gallons / minute.

Example 2

When a mobile moves in a straight line with constant speed, the slope of the position graph as a function of time is none other than said speed. The graph shows a mobile with positive velocity, which means that it is moving away from the origin.

References

1. Alvarez, J. The slope of a highway. Recovered from: geogebra.es.
2. Carena, M. 2019. Pre-University Mathematics Manual. National University of the Litoral.
3. Hoffman, J. Selection of Mathematics Topics. Volume 4.
4. Jiménez, R. 2008. Algebra. Prentice hall.
5. Stewart, J. 2006. Precalculus: Mathematics for Calculus. 5th. Edition. Cengage Learning.
6. Zill, D. 1984. Algebra and Trigonometry. Mcgraw hill.