Point-Slop form plays a crucial role in the study of the linear equations especially to build an equation of a straight line. The concept of Point-Slop form is very useful to understand the fundamental core dimensions of different fields like finance, banking, engineering, medicine and other science subjects.

In this article, we will elaborate the concept of Point-Slop form, its definition, formula, applications of Point-Slop form in real world context with solved examples.

## What is Point-Slope Form?

The Point-Slope form is a technique which is used to write or find a linear equation in two variables for a straight line whose slope and a point on the line is known and this line passes through this known point.

## Formula:

Mathematically, the Point-Slope form can be defined as

(y – y_{1}) = m (x – x_{1})

Where m is the gradient(slope) of the given line and (x_{1}, y_{1}) is the specific known point on the given line. Here it is important to note that (x, y) is a random point.

Let us know how the Point-Slope formula can be formed using the slope formula. We know that

m = 𝚫y/ 𝚫x =

m = (y_{2} – y_{1})/ (x_{2} – x_{1}) …………. (1)

To compute the slope of the line, here we use the known point which is (x_{1}, y_{1}) and the random point (x, y). Therefore, equation (1) will become

m = (y – y_{1})/ (x – x_{1})

multiplying both sides by (x – x_{1})

m (x – x_{1}) = (y – y_{1})

Arrange the above equation.

(y – y_{1}) = m (x – x_{1}) which is the point-slope form of the linear equation.

## Applications of Point-Slope Form:

The Point-Slope form of a linear equation plays a vital role to obtain useful insights and to avail key information. Here we brief some of its important applications in the following.

### Sketch Graphs of Linear Equations:

Point-Slope form helps us to locate the graph an equation representing a line. If you know a piece of information about the slop and any point on the line you will be able to epitomize to find more points and locating them on the line.

### Economics & Finance:

Point-Slope form enables us to apprehend the economic and finance indicators like supply, demand, price, quantity etc. as well as to find out useful insights from their curves. We can make sense that in point-slope form, a given specific point denotes price of the supplying indicators and their quantity where slope elaborates their rate of change.

### Engineering:

In the field of engineering especially in civil department point-slope form is used frequently. For example in the construction of buildings to manage their designs and structures like declivity of stairs, roof, roads as well as to understand their angle of leaning (inclination).

### Data analysis & Statistics:

Linear regression is important and core dimension used statistics and during study of the data analysis. Point-Slope from is applied to get more precise understanding of variables which represent regression equation. So, point-slope form plays an exceptional role to explain relationships b/w variables and get more indubitable regression line.

### Navigation & GPS:

The concept of point-slope from is used in Navigation and GPS system as well. The position of the satellite and receiver can be anticipated with the help of proper understanding of point-slope form.

### Physics:

The point-slope form is vastly used in the linear relationships of different quantities like speed, velocity, acceleration to understand different object’s motion precisely. For example, in point-slope from specific point is used to locate the initial position of the moving objects or particles and slope can find out their velocity, acceleration etc.

## Examples:

Here we will solve some examples to understand precisely the concept of Point-Slope form.

**Example 1:**

Find out the point-slope equation if slope of the line is -2 and passes through the point (2, -3).

**Solution: **

**Step 1.**

Given information

m = 2, x = 2 and y = -3

**Step 2.** Write down the general form of the point-slope form of linear equations.

y – y_{1} = m (x – x_{1})

**Step 3.**Put the values in the general form of the point-slope form of linear equations.

y – (-3) = (-2) (x – 2)

y + 3 = -2 (x – 2) which is the point-slope equation of the straight line.

Use a point slope form calculator to get the results in seconds without performing any calculations.

**Example 2. **Determine the point-slope equation for a straight line passing through the points (2, 3) and (4, 6)

**Solution:**

**Step 1.**Given data:

Here x_{1} = 2, x_{2} = 4, y_{1} = 3 and y_{2} = 6

**Step 2.** First of all we find the slope of the straight line by using slope formula

m = (y_{2} – y_{1})/ (x_{2} – x_{1})

**Step 3.** Put the values to calculate slope of the straight line.

m = (6 -3)/ (4 – 2)

m = 3/2

**Step 4.**Write down the general form of the point-slope form of linear equations.

y – y_{1} = m (x – x_{1})

Here we can use anyone of the given point on the line (x_{1}, x_{2}). Let us use (4, 6)

**Step 5.**Put the values in the general form of the point-slope form of the linear equations.

y – 6 = 3/2 (x – 4) which is the point-slope form of the equation.

**Example 3.**Find out the point-slope equation for a straight line having slope m = ½ and passes through the point (-1, 4).

**Solution:**

**Step 1.**Given data:

m = 1/ 2, x_{1} = -1 and y_{1} = 4

**Step 2.**Write down the general form of the of point-slope form of the linear equations.

y – y_{1} = m (x – x_{1})

**Step 3.** Place the values in the above equation.

y – 4 = ½ (x – (-1))

y – 4 = ½ (x + 1) which is the required equation of the point-slope form of linear equation.

## Summary:

In this article we briefed the concept of Point-Slope form of linear equation. We discussed its definition, formula and how its formula can be formed using slope formula. Afterwards we elaborated its practical applications in various technical fields. In the last section, we solved some examples.