The point-slope form is particularly useful when you have a single point on a line and the slope of the line. It allows you to quickly write the equation of the line without having to use the y-intercept or other points on the line. It is also useful when working with real-world problems that involve rates of change or slopes.

Overall, the point-slope form is a powerful tool for analyzing and solving problems involving linear equations and is a fundamental concept in algebra and geometry. The point-slope form is a way of representing the equation of a straight line in a two-dimensional coordinate system.

It is called the point-slope form because it requires a point on the line and the slope of the line to write the equation in this form. In this article, we will discuss the definition of point-slope form, what is point-slope form, Derivation of point-slope form and also an example of the topic will be explained.

## Point Slope Form

The point-slope form is a method used to write the equation of a straight line, based on a known point on the line and the slope of the line. The point-slope form of a linear equation is expressed as y – y_{1} = m (x – x_{1}), where (x1, y1) represents the coordinates of the known point on the line and m represents the slope of the line.

This form is useful when you have a single point and the slope of the line, and you want to write an equation that describes the line passing through that point. The point-slope form of a linear equation can be transformed into other forms, such as the slope-intercept form or standard form, for further analysis or calculation.

The point-slope form is a way to write the equation of a straight line, given a point on the line and the slope of the line. It is also known as the slope-intercept form and is represented as:

The point-slope form is particularly useful when you know the slope of the line and a point on the line, but not the y-intercept.

The point-slope form is an essential concept in algebra and geometry, used to denote lines in various contexts, such as linear equations, graphs, and real-life uses. It allows us to quickly and easily write the equation of a line and make predictions about its behavior.

### Point Slope Formula

The point-slope-intercept technique can be used to determine a line’s equation. The equation of a line with a given point and a given slope is found using the point-slope form.

Only when the slope of the line and a spot on the line are known can this formula be applied. We also have other forms, like the slope-intercept form and the intercept form, which can be used to determine a line’s equation.

**(y – y _{1}) = m (x – x_{1})**

### Point Slope Formula Derivation

The point-slope formula for a linear equation can be derived using the slope formula and the point-slope form of a line.

The slope formula gives us the slope of a line passing through two points (x1, y1) and (x2, y2):

**m = (y _{2} – y_{1}) / (x_{2} – x_{1})**

The point-slope form of a line describes a line passing through a point (x1, y1) with slope m:

**y – y _{1} = m (x – x_{1})**

To derive the point-slope formula, we can start with the slope formula and simplify it by using one of the points on the line:

**m = (y _{2} – y_{1}) / (x_{2} – x_{1})**

using (x_{2}, y_{2}) = (x, y)

m = (y – y_{1}) / (x – x_{1})

Now we have the slope expressed in terms of the point (x, y) and the known point on the line (x1, y1). We can rearrange this equation to obtain the point-slope form of a line:

**(y – y _{1}) = m (x – x_{1})**

This is the point-slope formula, which gives the equation of a line passing through a point (x1, y1) with slope m.

### Note:

- The point slope form equation of a line with slope “m” and passing through the point (x
_{1}, y_{1}) is y – y_{1}= m (x – x_{1}). - A parallel line’s equation is of the form y = b when it passes through the points (a, b).
- A straight line that passes through (a, b) has an equation of the form x = a.
- In this unique situation, the point-slope form cannot be applied.

## Examples of point slope form

**Example:**

A line passing through the coordinates (4, -5) and having a slope of (-1/2) should have an equation.

**Solution:**

Given The point on the given line is: (x_{1}, y_{1}) = (4, -5)

The slope of the line is: m = (-1/2)

**Step 1:**

The equation of the line is found using the point-slope form: y – y_{1}= m (x – x_{1}). Now put the given value in the equation of the line

y − (−5) = (−1/2) (x − 4)

y + 5 = (−1/2) x – 4

**Step 2**

For simplification, Subtracting 5 from both sides,

y = (−1/2) x − 9

Thus, the equation of the required line is, y = (−1/2) x – 9

The equation of the line can be determined with the help of a point slope calculator to get the result in a couple of seconds.

## Summary

In this article, we have discussed the definition of point-slope form, the meaning of point-slope form, derivation of point-slope form, and also with the help of an example topic will be explained. After studying this article everyone can defend this topic.