Slope of Line

In Mathematics, the slope of gradient characterizes both the steepness and direction of a line. The letter ‘m’ is used to represent the slope.

The slope is always estimated by determining the ratio of the “vertical change” to the “horizontal change” between any two different points on a line. Sometimes, this ratio is expressed as a quotient (“rise over the run”) giving a similar number for every two distinct points on the same line. A line that is drawn on a plane, and is decreasing has a negative “rise”. The line may be functional as laid down by the road surveyor, or in the diagram that exhibits a road or roof either as a description or a plan.

Read on to have detailed information on the slope of a line.

Slope of Line Definition

As discussed above, the slope of a line is the ratio of the “vertical change” to the “horizontal change” or the ratio between the change of x and the change of y.

Sometimes, the slope is expressed as rise over run. You can determine slope by observing walking up the flight of stairs, dividing the vertical change, which comes in the numerator by the horizontal change, which comes in the denominator.

The slope, m, of a line, is defined as:

$m = \frac{\mathtt{change~~in~~y}}{\mathtt{ change~~in~~x}} = \frac{\vartriangle y}{\vartriangle x}$

What is the Slope Formula?

The slope of a line in the coordinate plane comprising both of x and y axes is usually denoted by the letter ‘m’ and is defined as the ratio of the change in y coordinate to the corresponding change in x coordinate, between two different points on a line. Accordingly, the slope of a line formula is given as :

$m =\frac{\vartriangle y}{\vartriangle x} =\frac{\mathtt{verticle ~~change}}{\mathtt{Horizontal ~~change}} =\frac{\text{rise}}{\text{run}}$

How to Measure the Steepness and Direction of a Line?

The steepness, grade, or incline of a line is measured by the absolute value of a slope. A slope with greater absolute values represents a steeper line. The direction of a line is either decreasing, increasing, vertical or horizontal.

• A line is increasing if it moves up from left to right. The slope is positive i.e. m > 0.

• A line is decreasing if it moves down from left to right. The slope is negative i.e. m < 0.

• The slope is zero if a line is horizontal.

Here, we can see that irrespective of the two points we choose, the value of the y-coordinate always remains the same; i.e it is always 3. Hence, the change in the y value along the line is zero. No matter what the change in x value along the line, the slope must always equal zero.

Slope (m) =$\frac{0}{\mathtt{change ~ in ~ x}}$=

0 divided by any number is always 0. Therefore, horizontal lines have a slope of 0.

• The slope is undefined if a line is vertical.

In this situation, irrespective of what two points we choose, the value of the x-coordinate always remains the same; it is always 2. Hence, the change in x value along the line is zero.

Slope (m) = $\frac{\mathtt{change ~~ in ~~y}}{0} $=

As we cannot divide by 0, the slope of a vertical line is infinite. Therefore, vertical lines have an infinite slope.

How to Find the Slope of A Line?

Given two coordinates $(x_{1},y_{1})$ and $(x_{2},y_{2})$ the change in variable x from one to another is $x_{2} -x_{1}$ (run) while the change in variable y from one to another is $y_{2} -y_{1}$ (rise). Substituting both the values in the above equation generates the following slope of a line formula:

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The above-given slope of a line equation is not valid for a vertical line, parallel to the y axis (refer to Division by Zero), where the slope can be considered as infinite, hence, the slope of a vertical line is considered undefined.

Let us understand how to find the slope of a line with an example.

Example 1:

Suppose a line runs through two points: A= (3,2), and B= (15,8). By dividing the difference in y- coordinates by the difference in x - coordinate, one can get the slope of the line as shown below:

$m=\frac{\vartriangle y}{\vartriangle x} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{8-2}{15-3} =\frac{6}{12} = \frac{1}{2}$

As the slope is positive, the direction of a line is increasing.

Also |m| < 1, the incline is not very steep (incline is less than $45^{\circ}$).

How to Find Slope from a Graph

Here in the below given figure, if the angle of inclination of the given line with the x-axis is $\theta$ then, the slope of the line is given by $\tan\theta$. Thus, there is a relation between the lines and angles.

Slope of a line is given as $m =\tan\theta$.

Here in the above image we can see that there are two points present on the line. The two points are A and B where A = $(x_{1},y_{1})$ and B= $(x_{2},y_{2})$ lie on the line with $( x_{1} \neq x_{2})$ then the slope of the line AB is given as:

$m =\tan\theta = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Where $\theta$ is the angle which the line AB makes with the positive direction of the x-axis. $\theta$ lies between $0^{\circ}$ and $180^{\circ}$

It must be noted that $\theta = 90^{\circ}$ is only possible when the line is parallel to the y-axis i.e. at $x_{1} = x_{2}$ at this particular angle the slope of the line is undefined.

Slope of Parallel Lines

Let us consider given lines i.e, $l_{1}$ and $l_{2}$ are parallel with inclinations α and β respectively. For two lines to be parallel their inclination must also be equal i.e. α=β. Thus this results in the fact that tan α = tan β. Therefore, the condition for two lines with inclinations α, β to be parallel is tan = tan β.

Slope of Perpendicular Lines

A set of perpendicular lines always has a 90º angle between them. Let us suppose we have two perpendicular lines$l_{1}$ and $l_{2}$ in the coordinate plane, inclined at angle and .

Suppose if they are perpendicular, we can say that β = α + 90°. (Using properties of angles)

Their slopes can be given as: $m_{1}$ = tan(α + 90°) and $m_{2}$ = tan .

$m_{1}= -\cot\alpha = \frac{-1}{\tan\alpha}$

Here we know that $tan\alpha = m_{2}$

Thus

$m_{1} = \dfrac{-1}{m_{2}}$

$m_{1} \times m_{2} = -1$

Thus, for two lines to be perpendicular the product of their slope must be equal to -1.

Slope for Collinearity

For two lines AB and BC to be collinear, the slopes of both lines must be equal, and they must pass through at least one point in common. Thus, for three points A, B, and C to be collinear the slopes of AB and BC must be equal.

When ab' c' = a' b' c = a'c'b, the two lines are collinear if their equations are ax + by + c = 0 and a' x+b' y+c' = 0.

Angle Between Two Lines

Suppose when two lines intersect at a point then the angle between them can be expressed in terms of their slopes and is given by the below formula:

$\tan\theta = |\frac{m_{2}-m_{1}}{1-m_{2}-m_{1}}|$,

Where $m_{1},m_{2}$are the slopes of the given lines AB and CD respectively.

Solved Problems

1. Find the slope of a line between the points P = (2, -1) and Q = (4,1).

Solution:

Given,the points P = (2, –1) and Q = (4,1).

As per the slope formula we know that,

Slope of a line, $m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$\Rightarrow m= \frac{1-(-1)}{4-2}$

$\Rightarrow \frac{2}{2}$

$\Rightarrow m=1$

2. Sowyma was checking the graph, and she realized that the raise was 20 units and the run was 5 units. What should be the slope of a line?

Solution:

Given that, Raise = 20 units

Run = 5 units.

We know that the slope of a line is defined as the ratio of raise to the run.

i.e. Slope, $m=\frac{\mathtt{rise}}{run}$

Hence, slope = $\dfrac{20}{5}$ = 4 units.

Therefore, the slope of a line is 4 units.

3. Find the slope of the straight line y-1=5x+2

Solution:

We must solve the equation for y.

y = 5x+2+1

y = 5x+3

y = 5x+2+1

y = 5x+3

If we look at the point-slope equation of a line

y=mx+b

y=mx+b, we can conclude that the slope of this straight line is m=5

Practise Questions

1. Find the slope of the line 2y-3x=5

a) $\dfrac{3}{2}$

b) $\dfrac{1}{2}$

c) $\dfrac{3}{4}$

d) $\dfrac{1}{3}$

2. Find the equation of the line that passes through the point (2,5) and its slope is m=3.

a) y=2x+1

b) y=3x-1

c) y=3x-2

d) y=2x-3

3) Find the slope of a straight line that passes through the points (1,3) and (-1,-1).

a) 3

b) 1

c) 2

d) 8

Answers

1) a

2) b

3) c

Conclusion

Here in this article we have learnt about the slope. The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. We also learned about the slope for different types of lines such as parallel lines,perpendicular lines. Along with these we also discussed slope concept for two more topics i.e, collinearity and angle between two lines. Hope this article has helped you!.