Slope of a Line

One of the main challenges students faces while learning maths is presenting ideas clearly and determining accurate results. Since mathematics involves a lot of concepts, this explains the coexistence of the surprising mix of struggle and competence. And one of such concepts that act as a nightmare to students is the ideas surrounding slope. While instructors think that it expresses a number that measures steepness, students often confuse them with a pair of numbers only separated by a slash. If you are also in the same dilemma and confuse yourself with the concepts, read on to gain clarity.

What Is The Slope Of A Line?

The slope of a line is often defined as the change in the 'y' coordinate concerning the change in the 'x' coordinate of that particular line. The ultimate change in the y coordinate is Δy, while the net change in the x coordinate is Δx. Thus the change can be denoted by:

m = Δy/Δx

here ‘m’ is the slope.

The slope of a line is the measure of the direction and steepness of the line. It can be calculated by using two distinct points lying on a particular line. Determining the slope of lines in a specific coordinate plane can enable you to predict whether the lines are perpendicular, parallel or none. Moreover, you can do this without using a compass.

How To Find Slope With Two Points?

The slope of a line can be calculated by using two points lying on the same straight line. The coordinates of the first point are represented by 'x1' and 'y!' respectively. While the second points are denoted by 'x2' and 'y2', respectively.

Let us look at an image to understand this better.

Let’s say the coordinates of the two points be,

P₁= (x₁ , x₂)

P₂= (x₂,y₂)

The equation of slope is:

m=(y2-y1)/(x2-x1)

As stated earlier, the slope is the change of the y coordinate concerning the x, so by putting the values of Δy and Δx in the equation of slope, we get:

Δy = y₂ – y₁

Δx = x₂ – x₁

Thus, these values in a ratio give,

Slope = m = tan θ = (y2 - y1)/(x2 - x1)

Here, m is the slope, and θ is the angle made by the line with the positive y-axis.

Example:

Question 1: Find out the slope of the line between the points (1,2)(1,2) and (4,5)(4,5).

Solution:

Let us consider (1,2) point 1 and (4,5) like point 2.

Thus, the coordinates will be x₁, y₁(1,2) and x₂, y₂(4,5)

By using the slope formula, we get: m=y₂−y₁/x₂−x₁

When you substitute the values in the slope formula, you get:

Y of the second point minus y of the first point: m=5-2/ x₂−x₁

X of the second point minus x of the first point: m= 5-2/4-1

When you simplify the denominator, you get: m=3/3

Therefore, m=1

The Slope Formula

The slope of a line formula is the ratio of the rise and the run or the rise divided by the run. It is used to describe the steepness of a line in the coordinate plane. Calculating the line slope equals summing the hill between two different points. Generally, to find the slope of a line, you need t have the values of two other coordinates in one line.

The formula is:

m= rise/run= y₂−y₁/x₂−x₁

where,

m is the slope

(x₁ , x₂) are the coordinates of the first point

(x₂,y₂) are the coordinates of the second point in the line

Example:

The slope is 1, passing through the point (-1 and -5). So what is the equation of the line?

Solution:

As given, the slope is 1, so the value of m will be 1.

In the general equation: y=mx+b and when we substitute the value of m, we are left with:

y=x+b

Now when we put the values onto the equation, we get b= -4

Hence when we substitute the value of m and b from the equation, we get our final result. The final equation is  y= x-4

Types Of Slope

A slope can be classified into 4 different types depending on the relationship between the two variables, x and y. This is so that you can determine the value of the slope or gradient of the line obtained.

• Positive Slope

A positive slope indicates the slope of the line that is inclined upwards when you are moving from left to right. The angle made by the line with a positive slope is acute, and it signifies that when x rises, so does y. A positive slope provides a directly proportional relationship between the two variables, x and y.

• Negative Slope

A negative slope illustrates that a line is downward while moving from left to right. Thus, the line’s rise to the run ratio is a negative value. Moreover, it goes to signify that when x increases, the value of y falls.

• Zero Slope

A line in zero slope is just a horizontal line parallel to the x-axis of the coordinate system. The rise for this particular slope is zero, and it makes an angle of 0 degrees or 180 degrees, with the positive direction being on the x-axis. So with a zero slope, any two points on the line have the same value for both x and y coordinates.

• Undefined Slope

An undefined slope is the slope of a vertical line. Here the x coordinates do not change regardless of what the candidates are. In this type of slope, the vertical lines rise straight up and fall straight down, and they never turn left or right. The slope of such a vertical line is undefined.

Distinguish Between Graphs Of Lines With Negative And Positive Slopes.

The slope is an essential concept to learn when dealing with graphs and lines. Knowing the type of slope will only make it easier to find the answers. Now, since there are many different types of slopes, it is tricky to understand their differences. If you are also wondering about the difference between a negative and positive slope, take a look.

A positive slope has increment in both and x and y lines simultaneously. So visually, the line moves up when you go from left to right.

However, in a negative slope, as x increases, y decreases, and visually, the line moves down as you go from left to right.

Let us look at some examples to understand this better.

Let's say the line y= 3.5x+4 has a positive slope. Since m here is 3.5, it is positive and a positive y-intercept is b=4. This means that the line passes through a point (0,4) where it is the y-intercept. To determine the intercept, we need to set y=0 and solve

• y = 3.5x + 4 [line equation]
• 0 = 3.5x + 4 [set y to 0 to find x-intercept]
• -3.5x = 4
• x = -8/7

So, the x interception point is -8/7,0. The slope of 3.5 can also be written as rise/run=3.5/1=7/2. This tells us that every time y increases by 7, x increases by 2.

Let’s say the line y=8x-16. It has a positive slope where m=8 and a negative y-intercept where b=-16. This means that the line passes through a point of 0, -16 where it meets the y-intercept. To determine the intercept, we need to set y=0 and solve

• y = 8x – 16 [line equation]
• 0 = 8x – 16 [set y to 0 to find x-intercept]
• -8x = -16
• x = 2

Thus, the x-intercept is point 2,0. The slope 8 can also be written as rise/run=8/1. It tells us that when y increases by 8, x increases by 1.

Let’s say the line y=-5x has a negative slope since m+-5, which is harmful and has a zero-y-intercept which means b=0. This means that the line passes through the origin with coordinates as 0,0. The origin is both the x and the y-intercept for the particular line. The slope can be written as rise/run=-5/1. This enables you to understand that y decreases every time by 5 and x increases by 1.

Most Frequently Questions Asked By Students

1. How Do You Calculate The Slope Of A Hill?

Ans: Calculating the slope of a hill is not a daunting task as you can convert the rise and run to the given units, and they divide the rise by the run. Additionally, multiply the number by 100, and you have the slope for your hill. For instance, a slope of 0.5 would mean a rise of 1 unit for every 2 units horizontal and a slope of 2.0 would be a rise in every 2 units horizontal.

2. How To Find Slope Of An Equation?

Ans: Follow these 3 ways to determine the slope of an equation:

• Find the slope of a linear equation with the formula y=mx+b
• Find the slope of two points with the formula m=y₂−y₁/x₂−x₁
• Use differential calculus to find the slope of a curve.

3. How Do You Find The Slope Of A Curve?

Ans: To find the slope of a curve, follow these steps:

• Differentiate the curve's equation and first find out the derivative of the curve. It is usually denoted by dy/dx.
• After differentiating, you will get the equation of the slope.
• Place the value of x in the equation and find out the answer for the slope.