What is meant by Slope?

The steepness of a line or hill is known as its slope. It is the ratio of the change in vertical distance to the change in horizontal distance. It is generally represented by the letter ‘$m$’.

Interpretation of Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

It is possible to determine the direction of a line based on a sign as well as the value of $m$:

When $m$> 0, the line is increasing in an upward direction.

When $m$< 0, the line decreases in the downward direction.

When $m$= 0, the line is horizontal.

Positive, Negative, and Zero Slopes

When two variables are positively related that means when x value increases y also increases and x decreases y also decreases. Graphically when the line of the graph moves up from the left to the right side then the line rises this is known as a positive slope. 

Refer to the following graph:

A graphically negative slope is estimated when the line of the graph moves down from left to right side, which results in the falling of the line.

Refer to the following graph:

When the line is horizontal, i.e., the $y$-component remains the same despite any change in the $x$ component, then it gives a zero slope value.

Steps to Calculate the Slope

Let’s check the following listed steps:

Step 1: Find out any two points on the line.

Step 2: Determine the value of the vertical distance in units up or down. It is nothing but your rise value. It is a numerator value.

Step 3: Estimate the horizontal distance from one point to the other, i.e., measure the distance between the left point and the right point. It is nothing but the run value. It is a denominator.

Step 4: Take the ratio of the rise and the run which results in the value of the slope.

The Slope of a Line

The slope of a line can be calculated using different methods depending on the given information.

The Slope of a Line Passing Through Two Points

It is easier to find out the slope of a line when two points are given with the graph line. In this case, you need to use the grid paper on which you have to plot the point and make a line then count the value of the rise and the run. But what if you don’t graph the line image to count the run and the rise value? Instead of doing this extra time and energy-wasting method, you can calculate the value of the slope of a line without graphing method. Let’s check the method in detail:

Here are two points that make a line. So use subscripts to distinguish or name the points of a line as:

First point: $\left(x_1,y_1\right)$ Read the first point as “$x$ sub 1 and $y$ sub 1”

Second point: $\left(x_2,y_2\right)$ Read the first point as “$x$ sub 2 and $y$ sub 2”

If there are more points like a third, fourth, etc then named them $\left(x_3,y_3\right),\left(x_4,y_4\right)$ and so on.

So, you can use the formula to calculate the slope of a line between two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ as:

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

That means the slope of a line is the ratio of change in y-coordinates and x-coordinates of the line. The change in y-coordinates is a subtraction between the value of the y-coordinate of the second point and the value of the y-coordinate of the first point. The change in x-coordinates is a subtraction between the value of the x-coordinate of the second point and the value of the x-coordinate of the first point.

For example,

Find the value of the slope of a line through the points $\left(2,3\right)$ and $\left(7,4\right)$.

Solution:

Name the given points of the line as:

Start point: $\left(x_1,y_1\right)$= $\left(2,3\right)$

Endpoint: $\left(x_2,y_2\right)$ = $\left(7,4\right)$

Use the following formula:

Put the given points in the above formula,

Gradient or Slope of a Line

Consider, the angle made by line L with the x-axis is θ and the gradient value is m then the slope of the line formula is given by the following expression:

Refer to the following graph:

Case 1: $x_2\ne x_1$

If a line L passes the through points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ such that $x_2\ne x_1$, then the slope of the line L is expressed as:

Where angle made by line L with x-axis in the positive direction is $\theta$ which lies between $0°$ and $180°$.

Case 2: Undefined slope: $x_2=x_1$

When line L is parallel to the y-axis such that $x_2=x_1$ that means the angle made by line L with the x-axis is $90°$ then the slope of the line is undefined.

Equation of a Line

Following are the different forms of equation of a line:

Point-Slope Equation

The equation of a line passing through $\left(x_1,y_1\right)$ with slope $m$, is given by the following formula:

$y-y_1=m\left(x-x_1\right)$

Where,

Coordinates of the first point: $\left(x_1,y_1\right)$

The slope of the line: $m$

The point-slope equation emphasizes the slope between line and point which is not any intercept (x-intercept or y-intercept).

Slope-Intercept Form

Straight-line equations are linear equations that have variable but simple expressions with terms. Sometimes a line is given with the slope value and intercepts value on one of the axes of the graph. In this, case you need to the following guidelines to calculate the slope equation of the given line.

Case 1:

Consider, from the origin the line L with slope value $m$ cuts at a distance $c$ on the y-axis. So, here the distance c is called the y-intercept of line L. The coordinates of the point where the line meets the y-axis are $\left(0,c\right)$. That means line L has slope m and it passes through a fixed point $\left(0,c\right)$. Then the slope-intercept form of a line L is given by the following equation:

Thus, the point $\left(x,y\right)$ lies on line L which has slope m and y-intercept c if and only if $y=mx+c$.

For example,

Find the slope-intercept form of slope 2 and y-intercept 3.

Solution:

We know that the slope is represented by m and the y-intercept is c. Then the given data is expressed as:

m = 2

c = 3

Calculate the slope-intercept form as:

$y=mx+c$

Put the given values in the above equation,

$\begin{array}{l}y=2x+3\\ y-2x-3=0\end{array}$

Thus, the slope-intercept form or equation is derived as $y=2x+3$ or $y-2x-3=0$.

Case 2:

Consider, from the origin the line L with slope value m cuts at a distance d on the x-axis. The distance d is called the x-intercept of line L. The coordinates of the point where the line meets the x-axis are $\left(d,0\right)$ . That means line L has slope m and it passes through a fixed point $\left(d,0\right)$. Then the slope-intercept form of a line L is given by the following equation:

Thus, the point (x, y) is said to lie on line L if and only if $y=m\left(x-d\right)$.

For example,

Find the slope-intercept form of slope 4 and x-intercept 2.

Solution:

We know that the slope is represented by m and the x-intercept is d. Then the given data is expressed as:

m = 4

d = 2

Calculate the slope-intercept form as:

Put the given values in the above equation,

Thus, the slope-intercept form or equation is derived as $y=4x-8$ or $y-4x+8=0$

Common Mistakes

Divide each term by the coefficient on y during the identification of y-intercept and slope by using the slope-intercept formula. Once you have isolated y values you can easily identify the values of slope and y-intercept.

Remember that vertical change or change in y-coordinates is the numerator of the slope formula. Do not place it in the denominator in the formula.

The number will be positive if you count up or right and if you count down or left then it will be a negative number.

The value of c is either positive or negative which depends on the intercept made on the positive or negative side of the y-axis.

Formula

Context and Application

In real-world relationships such as unit cost, speed, and rates of changes are easily interpreted with the help of the slope equation.

Nowadays, there are many online slope calculators available in which you need to put the value of the variable and the slope is easily estimated.