The point slope form equation describes the relationship between the variables x and y on a line. By specifying a point on the line and its slope, we can uniquely determine the equation of the line.

It is a powerful method for representing linear relationships and is often used in various fields such as mathematics, physics, engineering, and economics. It provides a concise and efficient way to describe a line based on a single point and its slope, making it a valuable concept in algebraic representation and analysis.

In this article, we have discussed the definition of point-slope form, formula, linear equation, derivation, and application of point-slope form. Also, with the help of examples, the topic will be explained. After a brief study of this article, anyone can defend this topic easily.

**Point Slope Form**

It is used to find the equation of line in slope intercept form, for a given set of points and slope. The points are said to be the coordinate’s points of the line and the slope is the measure of the steepness of the line.

The slope, denoted by ‘m’, measures how steep the line is. It indicates the rate at which the y variable changes for the x variable. A positive slope indicates an upward slant, while a negative slope indicates a **downward slant**. A slope of zero corresponds to a horizontal line, and an undefined slope corresponds to a vertical line.

**Point Slope Form Equation**

In point-slope form, an equation of a line is written as:

y – y₁ = m (x – x₁)

- Here x
_{1}and_{, }y_{1}indicate the coordinates. - ‘m’ shows the slope of the line.
- (x, y) show the variable.

You can find the equation of the line in the form of the above point slope equation with the help of a point slope form calculator. It will allow you to find the equation of the line either with the help of coordinate points of the line or with the help of given slope and point of the line.

**Point Slope Form: Linear Equation**

The linear equation of point-slope form is given by:

y – y_{1} = m (x – x_{1})

where:

- (x
_{1}, y_{1}) show the point on the line - m shows the slope of the line

In this form, you can substitute the values of the point (x_{1}, y_{1}) and the slope (m) to write the equation of a line.

**Derivation**

To derive the point-slope form of a line, we start with the general equation of a line, which is the slope-intercept form:

y = mx + b

- m represents the slope
- b represents the y-intercept

Now, let’s (x_{1}, y_{1}) denote the point on the line. Now we find the slope of a line passing through the point:

- First, let’s calculate the slope (m) of the line using the given point and the slope formula:

m = (y – y_{1}) / (x – x_{1})

- Next, we substitute this value of m into the slope-intercept form:

y = m(x) + b

- Replacing m with the value calculated above, we have:

y = ((y – y_{1}) / (x – x_{1})) × (x + b)

- Simply the given equation:

y = (y – y_{1}) × (1 / (x – x_{1})) × (x + b)

y = (y – y_{1}) × (x / (x – x_{1}) + b)

- In the last step we rearrange the equation and put the given value in the equation:

y – y_{1} = (x – x_{1}) × ((y – y_{1}) / (x – x_{1}))

- Simplifying further, we get:

y – y_{1} = m (x – x_{1})

This demonstrates the line’s equation in the point-slope form, where (x_{1}, y_{1}) is a line point and m is the line’s slope.

Here, (x_{1}, y_{1}) is show the point on the line, and also, m represents the slope of the line.

## Point Slope Form: Applications

The point-slope form of a linear equation (y – y_{1} = m (x – x_{1})) has several applications in mathematics and real-world scenarios. Here are a few examples:

**Finding the equation of a line:**

Given a point on a line and its slope, you can use the point-slope form to quickly write the equation of that line. This form provides a straightforward way to describe the relationship between x and y for all points on the line.

**Graphing a line:**

It helps in graphing a line efficiently. By identifying a point on the line and its slope, you can plot that point and use the slope to find additional points to create the line on a graph.

**Linear interpolation:**

In data analysis or curve fitting, the point-slope form can be used to estimate values between known data points. By selecting two neighboring points and assuming a linear relationship, you can use the form to find intermediate values.

**Slope comparison:**

The point-slope form allows for a straightforward comparison of slopes between different lines. By comparing the values of m, you can determine which line is steeper or flatter.

**Problem-solving in physics:**

The point-slope form is used to solve various physics problems involving motion, such as determining the position, velocity, or acceleration of an object along a straight path.

**Calculating rates of change:**

Since the slope represents the rate of change, the point-slope form can be used to calculate rates of change in different contexts. For example, in economics,it can be used to analyze production costs, revenue, or demand elasticity.

**Examples of Point Slope Form**

**Example 1:**

Evaluate the equation of a line whose slope is 3 and passes through the point (2, 5).

**Solution:**

Given: Point (2, 5) and slope (m) = 3.

Step1:

We’ll use the point-slope form of the line equation:

y – y_{1} = m (x – x_{1})

Substituting the given values into the equation:

y – 5 = 3(x – 2)

Step 2:

Expanding the right side:

y – 5 = 3x – 6

Step 3:

Next, we isolate the y-term by adding 5 to both sides:

y = 3x – 6 + 5

Simplifying further:

**y = 3x – 1**

**Example 2:**

A line passing through the coordinates (9, -4) and having a slope of (-1/5) should have an equation.

**Solution: **

m = (-1/5)

Step 1:

The point-slope form is used to determine the line’s equation: y – y_{1}= m (x – x_{1}).

Now put the given value in the equation of the line

y − (−4) = (−1/5) (x − 9)

y + 4 = (−1/5) x – 9

Step 2:

For simplification,

Subtracting 4 from both sides,

y = (−1/5) x − 13

Thus, the equation of the required line is, **y = (−1/5) x – 13**