Knowing how to find or calculate the slope of a line can be applied on more occasions than you think. In addition, the slope of a line is a key knowledge base for many subjects. For example, subjects such as algebra, mathematics applied to social sciences, economics…

Knowing the slope of a straight line is knowledge that the vast majority acquire during their time at school. But, as often happens, this knowledge is often forgotten over time. Or, it may also happen that we do not understand the teacher’s explanations well. Or, that it is a concept that chokes us and we do not know how to solve it.

Although, on the other hand, the vision of a slope is constant in our day to day. What happens is that surely we do not think about it when we see them. For example, we can think of a car going down a ramp or a person going down the stairs. In the same way, we can see how an animal climbs a hill or how mountains are scaled. These are examples of how present the slopes are in our lives.

Well, in this article we intend to teach the basic rules to know how to calculate the slope of a line.

What is a slope?

Before knowing how to calculate the slope of a line, we must first know and understand what a line itself is. A slope is known in mathematics and sciences of social applications as the inclination of an ideal element, both natural and constructive, with respect to a horizontal straight line. In more technical words, the slope of a line is the tangent of the angle that gives rise to the line with the direction of the X axis (or abscissa axis).

All this information can be very confusing at first. Therefore, a slope can be defined in a more colloquial and simple way. We can say that a slope is the transformation that takes place on the Y axis (or the ordinate axis), divided by the transformation that the X axis (or abscissa axis) undergoes.

Before we know how to find the slope of a line, we need to know something. And it is something fundamental for the purpose of this exercise. In order to know a slope, the lines have to be straight. We cannot find the slope of a line if the line is curved. It seems obvious if we look at the name of the exercise, but many times, the clearer something seems to be, the less we understand it.

Characteristics of a slope

On the other hand, it is possible to think that the line, according to its inclination, can go up or down. Or, paying attention to technical names, the slope of a line can be both positive and negative.  As we can see, there are several types of lines. When a line is represented graphically, it is much easier to determine what type of slope it is. Since the inclination of the slope is what identifies if it is positive or negative.

Slope of a positive line

We say that a slope is positive when the line is increasing. This means that it starts at a low point and works its way up as you incline. In other words, we can speak of a positive slope when as the values ​​of x increase, the values ​​of y also increase.

Slope of a negative line

Contrary to the case we have just seen, we speak of a negative slope when the line is decreasing. That is, it starts at a high point on the chart and works its way down as you move up the chart. We can say, in another way, that a slope is negative if when the values ​​of x increase, the values ​​of y decrease.

Slope of a null line

The last case that can occur is this, that the slope of a line is absolutely zero. As we can imagine, the slope with zero value is a horizontal line. Or what is the same, this slope has no inclination.

How to find the slope of a line?

To calculate the slope of a line, we recommend that you get hold of paper and pencil. We can make our own graph to see in a much clearer way, how to calculate the slope. We can do this especially at the beginning, which is more likely to be difficult for us to understand if we still do not have practice.

In any case, you can find the slope of a line without a graph. But for this it is mandatory that we know two coordinates of the line that we intend to calculate. Each coordinate corresponds to a specific point on the line. For each point on the line, we can obtain two coordinates. One for the X value, which tells us where that point is with respect to the horizontal axis. On the other hand, we will also obtain the coordinate for the Y value, which clarifies where that point is with respect to the vertical axis.

We draw a graph and the line

The first step that we are going to take to calculate the slope of a line is to draw the graph that will serve as a guide. To do so, we will have to make two lines perpendicular to each other. To make it easier for you to understand it, you must make two lines that intersect in the center, as if it were a cross. Now we can see that we have four quadrants in total.

The first quadrant I is the one that corresponds to the upper right square. The second quadrant II, corresponds to the square on the upper left. Next, the third quadrant III, is located in the lower left square. Finally, the fourth quadrant IV is the one in the lower right square.

Now we have made the graph and, in addition, we can contrast the previous information about the inclination of the lines and see what a line with a positive, negative or zero slope would look like. The only thing left to do is draw the line from which we will later have to calculate its slope. We can make the line as we want, both ascending and descending. The only thing we can’t do is a zero line. Because, as is logical, if the line is zero it has no inclination, and if it has no inclination, there is no slope to calculate.

The important thing when making the straight line is that it be done with a ruler, so that it is effectively straight and does not have any undulations.

Calculate the slope of our line

Now it’s time to calculate the slope of the line we just drew. The first step we must take is to select the two points we need on the line. It doesn’t matter how high up the line the two points are. And it is not important in which quadrant the points you have selected are located.

Well, once the points have been chosen, we must determine the coordinates of each one. This is very simple, we just have to see the value it has on the horizontal axis and on the vertical axis. It doesn’t matter if the numbers are negative. Now choose one of the points with its coordinates already determined and we will give it the name x1,y1.  In this way, that point that we have chosen will be the one with the dominant coordinates.  And the other remaining point will be named x2,y2. So far everything has been very simple, and best of all, it will continue to be simple. All we have to do is apply the following formula.

m= y2 – y1/ x2 – x1

In the equation, m is the name assigned to the slope of a line and the other values ​​we have already obtained before. Therefore, it only remains to apply the formula.

Practical example

Now we are going to do a practical example in case there are any doubts that can be resolved. First we draw the graph and make the line. We should have something like this:

Next, we choose two points at random, which will be different from those of the drawing. In this case, the first point is in quadrant III, with coordinates of (-2, -3). While the second point is located in quadrant I, with coordinates of (3,4). In the parentheses, when putting the coordinates, the first number that is put is the one that corresponds to the X axis. Therefore, in the second place, the value of the Y axis is put. We have chosen the coordinates of the quadrant III as x1,y1 and those of quadrant I as x2y2.

Now we are going to apply the formula. In this case it would look like this: m= 4 – (-3)/ 3 – (-2). First, we obtain the result of the dividend and the divisor separately, without doing any further calculations. In our case it would look like this: m= 7/ 5. Finally, we only have to calculate the remaining division. Here it would give us a result of 1.4.

Finally, we can determine that our slope is 1.4 and, furthermore, we can say that it has a positive slope.

As we have seen, it has not been so complicated to determine the slope of a line.  Following these simple steps we have the way to calculate the slope of any line.  We hope that the information contained in this article has been of help to you in solving this mathematical problem and that, from now on, there will be no pending problems for you.