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Point Slope Form: Introduction, Rules, Examples, and Applications

By Mohsin Ali Last updated : October 20, 2023

The point-slope form is important in mathematics for figuring out the equation of a straight line that must pass through a specific point and is inclined at a specific angle to the x-axis (a point on the x-axis). Additionally, it facilitates our understanding of how each point on the line satisfies the equation describing it.

Since the slope of the line and its point are both known, the Point slope form is useful. The formula for the point-slope form and its detailed discussion shall follow.

What is Point Slope Form?

One definition of point slope form is an equation of a line expressed using only one point on the line and the line's slope. The point-slope form of the equation is shown as follows:

[y – y1] = m [x – x1 ]

Where,

• m is denoted as the slope of the line
• (x, y) is a random point on the line(which should be kept as a variable while applying the formula).
• (x_1, y₁) is a fixed point on a line

How to derive a point slope formula? (Derivation)

Let's use the method for calculating a line's slope to derive the formula of point slope form (prove the formula of point slope form). Consider a line whose slope is represented by the letter "m." Let's assume that the line's (x₁, y₁) known point is on it. Any point on the line, whose coordinates are unknown, may be used as (x, y).

As we know that

Slope (m) = (Difference in y-coordinates) / (Difference in x-coordinates)  
m = (y– y1) /( x – x1)
m (x – x1)  = y– y1
⇒y– y1   = m(x – x1)

How to Solve Point Slope Form?

To solve point-slope form for a given straight line to find the required equation of a given line, we have to follow the following steps,

• Step 1: List out the slope, 'm' of the straight line, and the coordinates(x₁, y₁) of the given point that lies on the line.
• Step 2: Substitute the given values in its formula: y - y₁ = m(x– x₁).
• Step 3: Solve to obtain the equation of the line in standard form.

Examples:

Example 1:

Find the equation of a line that passes through a point (0, -3) and whose slope is (-6).

Solution:

Point on the given line (x_1, y₁) = (0, -3)

m = -6

We know the point-slope formula

y– y₁   = m(x – x₁) 
⇒ y – (-3)    = (-6) (x-0)
⇒ y +3    =   –6x
⇒ y =   – 6x –3
⇒ y = – 6x –3
⇒ y = (–6) x – 3

Example 2:

Calculate an equation of a horizontal line that passes through a point (-3, 2).

Solution:

We know that

The slope of a line parallel to the x-axis is m = 0

And

The line passes through the point (x_1, y₁) = (-3, 2)

Using a formula of point slope form

y– y₁   = m(x – x₁) 
y − 2 = 0(x – (-3))
y − 2 = 0(x +3)
y − 2 = 0
y = 2

Example 3:

Find the equation of a line that passes through two points (1, 3) and (2, 0) using the point-slope form.

Solution:

The given two points are:

(x1, y1) = (1, 3)

(x2, y2) = (2, 0)

To utilize the point-slope form, first of all, we have to find the slope of the line.

The slope of the given line is obtained as follows:

m = [y₂ – y₁]/[x₂ – x₁] = (0 − 3)/(2 −1) = −3/1 = −3

⇒ m= −3

Now, apply the point-slope formula:

y– y₁   = m(x – x₁) 
y – 3 = (– 3) (x – 1)
⇒ y – 3 = –3x+3
y= –3x+3+3
y= –3x+6
y= (–3) x + 6

Applications:

The terminology of point slope form is widely used in our daily life in many departments. Some applications are mentioned below:

• In construction works like building roads, we must have to figure out how steep the road will be. It is meant to make the road safer and more effective; we must have to know the concept of slope. Similarly, when building stairs, one must have to know the concept of the slope so they are not too steep to walk on
• For skydivers/skiers/snowboarders, one should have to know knowledge of the slopes of hills to judge the dangers, speeds, etc.
• In production cells, like constructing wheelchair ramps and wheelbarrows, the idea or concept of slope is a major consideration. The reason is that in these machines all the measuring steps have much importance.
• In gaming, a game animator chooses many characters to make it interesting for the users. The reason is that whenever he designs a character to perform a specific action, such as a jump he has to move the character in such a way that it can move a few spaces up and a few spaces forward, etc.
• In art and drawing, the slopes of the lines drawn must be known to decide what would be the most artful or gorgeous way of looking an artist presents through his skills so that his art becomes attractive to all art lovers.
• In Surveying and Mapping, the Point-slope form is used to calculate gradients, slopes, and elevations. This information is crucial for designing roads, determining water flow, and creating topographic maps

Summary:

In the above article on Point Slope form, we acknowledged the concept of point slope form through its introduction, derivation of its formula, examples, and everyday applications as well. Its different kinds of solved examples can help us to solve all problems relevant to it.