Point Slope Form of a Line

The point slope form of a line helps find the exact equation of a straight line that can be inclined on the x-axis and passes through a specific point on the line. An equation should satisfy all the points on the line, and therefore, various methods are used to fulfill the requirements of the equation. Some of the important methods of finding an equation of a straight line are: 

• Point Slope Form of a Line
• Intercept Form
• Slope-Intercept Form
• Two-Point Form

Point slope form of a line is used only when the slope and the point of a line are given. With this information, one can derive the equation of a straight line. Let’s discuss the point slope form of a line and learn its formula. 

Point Slope Form of a line

The equation of a line satisfies every point on a line; if a point does not satisfy the equation, then the point is not on the specified line. The point slope form is used to find the representation of the equation of a line that passes through the specific point and makes an angle with the X-axis. The equation of a line is represented through various methods, and one of the methods is the point slope form of a line. 

It represents a straight line with the help of a slope and any point on the line. In short, it means that if a slope is m and the line passes through (x1, y1), the equation would be derived using the point slope form of a line.  

Point slope form of a line formula 

This point slope form of a line is used to find the equation of a line. Therefore, the line equation with a specified slope and a given point is important to find the point slope form. 

Formula: y–y1 = m (x–x1) 

where, 

• (x, y) is a random point on a line (it would be kept as variables when we would apply it in the formula)
• (x1, y1) is a fixed, determined point in a line
• m is a slope of the line

How to derive the point slope form of a line formula?

Let’s consider m as the slope of a line and  x1, y1 as the specific point of a line, and x and y as the random points on a line. 

We have a formula for slope which is as follows: 

Slope or m = (y2–y1)(x2–x1)

Multiplying both the sides by (x–x1),

m (x–x1) = y–y1. 

It can also be written as 

Y–y1 = m (x–x1)

Hence, the formula is derived from this method. 

Point slope form of a line examples 

So, let’s understand how to find an equation using the point slope form of a line formula: 

Case 1: 

Find the equation of a line passing from Point (3,-2) and having a slope of 4. 

Solution: Let (3,–2) ≡ (x1, y1) and 

Slope m = 4 

Therefore, by using Point Slope Form, the equation of the given line can be deduced from the formula 

y–y1 = m (x–x1)

∴ y–(–2) = 4 (x–3) (Substituting the values of x1, y1, and m) 

∴ y+2 = 4(x–3)

∴ y+2 = 4x–12

Rearranging terms of the equation,

y–4x= –12–2

∴ y–4x = –14

Implies,

4x – y = 14

The equation of the line passing from Point (3,-2) and having a slope of 4 is 4x – y = 14. 

So, this is how the equation of a line is obtained when a point and slope are provided in the question. 

Case 2: 

When a slope is not given, but the angle prescribed by it is given. 

Find the line equation if an angle formed by it with the x-axis is 30° and passes through (4, -3). 

Solution: 

Here, the slope of the line is not provided, but the angle formed by it is given. 

∴ By using the formula to find the slope

m = tan θ

∴ m = tan (30°)

∴ m = 1/√3

Given, (4, –3) ≡ (x1, y1)

Now using the formula of point slope form of a line, 

y–y1 = m (x–x1)

∴ y–(–3) = 1/√3(x–4)

∴ y+3 = 1/√3(x–4)

∴√3(y+3) = x–4

∴√3y + 3√3 = x–4

∴ x–√3y = 3√3 + 4

The equation of a line, if an angle formed by it with the x-axis is 30° and it passes from (4, -3), is 

 x – √3y = 3√3 + 4.

So, this is how we can find the line equation if the slope is not directly given, i.e., the angle formed by the line is given.

Case 3:

When neither the slope nor the angle is given, any two distinct points on the line are shown in the problem.

Find the line equation if the given line passes from points (2,5) and (4,-3).

Solution:

Let us assume, (2,5) ≡ (x1,y1) and (4,-3) ≡ (x2,y2)

In this case, we will find the slope of the line, m, by the formula,

m = (y2–y1) / (x2–x1)

∴ m= (–3–5) / (4–2)

∴ m= –8/2

∴ m= –4 ………………….(i)

Using point slope form of line,

y–y1= m(x–x1)

∴ y–5 = –4(x–2)

∴ y–5 = –4x + 8

∴ 4x + y = 8 + 5

∴ 4x + y = 13

The equation of the line, if the given line passes from points (2,5) and (4,–3), is 4x + y = 13.

These are some conditions that can be applied in different cases, and one can use the point slope form of a line to derive the equation of a straight line. You can either use these methods or the point slope form of a line calculator that can help to find an equation directly. Just input the coordinates and slope and find an answer. 

Conclusion

The point slope form of a line is thus used to find an equation of a straight line. The above-mentioned circumstances can consider the point slope form of a line formula to fulfill the requirements of an equation. So, if any of the conditions are provided to you in the form of a slope, point of the line or an equation, you can derive the nature of a straight line.