What is Point-slope form? Explained with Examples

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point slope form

A point-slope form is a mathematical toolfor solving the equation of a straight line by using a given point and slope. To delve into point-slope form, you need to know the equation of a straight line. The mathematical equation for a straight line can be expressed asax + by + c = 0. Here a, b, and c represent numerical values.

There are several methods to find the equation of a straight line. The technique chosen to solve a problem depends on the data provided. Some commonly used methods are:

  • Point-Slope form
  • Two-Point slope form
  • Slope-Intercept form
  • Intercept form

Our focus in this article will be directed toward the point-slope form. We will discuss the point-slope form in detail with its formula. Then we will discuss where the point-slope form originated. We will learnthe skill of determining the equation of a lineusing the point-slope form.

How can we describe the Point-slope form of a straight line?

Point-slope form of a straight line is an equation that expresses a line that passes through a point (x1, y1) with slope (m). Serval methods are used to determine the equation of the line. However, the point-slope form is the best tool for finding the equation of the line when the slope and point are provided.

The point slope form equation can be expressed mathematically as follows:

y – y1 = m (x – x1)

Point slope form formula

The formula of the point-slope form is given below:

y – y1 = m (x – x1)

Here,

  • m is the slope of the line
  • (x1, y1) are a given point that exists on the line
  • (x, y)is the coordinate of any other point on the line

Where the point-slope form originated?

The equation of point-slope form originated from the definition of slope. Suppose a line passes through a given point (x1, y1) with slope (m), and (x, y) is the coordinate of any other point on this line. The slope can be described as:

Slope = m = net change in y-coordinate / net change in x-coordinate

m = y – y1 / x – x1

When we multiply both sides of an equation by (x – x1), we get

y – y1 = m (x – x1)

That is the standard equation of point slope form.

The method for resolving the point-slope form

Here are the steps to resolve the point-slope form of a straight line:

  1. Identify the specific point (x1, y1) and slope (m) of the line.
  2. Write down the point-slope form equation that is y – y1 = m (x – x1)
  3. Substitute the given value in it.
  4. Simplify the equation by performing arithmetic operations.

Key points to consider about the Point-slope from

Here are some important points to remember about point-slope form:

  • An equation of Point slope form of the line that passes through the point (x1, y1) with slope ‘m’ is y – y1 = m (x – x1)
  • Slope intercept form y = mx + b is derived from the point-sloe form. In slope-intercept form, b represents the y-intercept.
  • If the line passes through the origin, in this case, (x1, y1) = (0, 0). The equation of point slope form becomes y = mx.

Solved Examples of Point-Slope Form

Let us determine the equation of a straight line by using the point-slope form in the following examples.

Example 1:

A line passes through (3, 5), with slope 4. Find the equation of this line.

Solution:

Step 1: Identify the point (x1, y1) and slope (m) of the line to find the equation of point slope form. Here,
x1 = 3, y1 = 5 and m = 4.

Step 2: Write down the equation of point slope form. That is y – y1 = m (x – x1)

Step 3: Substitute the given value in it.

y – 5 = 4 (x – 3)

Step 4: Simplify the above equation by performing arithmetic operations.

y – 5 = 4x – 12

4x –12 – y + 5 = 0

4x – y – 7 = 0

Hence, 4x – y – 7 = 0, is the equation of the line that passes through (3, 5), with slope 4.

You can take assistance from a point slope calculator to find the equation of line through point slope form by placing the value of slope and the coordinate points of the line.

Example 2:

Find the equation of the straight line that passes through (2, 3), which has a slope of 4 / 5.

Solution:

Here, (x1, y1) = (2, 3) and m = 4 / 5

An equation of point slope form is y – y1 = m (x – x1)

Put the given values in the equation of point-slope form, and we get

y – 3= (4 / 5) (x – 2)

After multiplying 5 on both sides of the equation

5y – 15 = 4 (x – 2)
5y – 15 = 4x – 8
4x – 8 – 5y + 15 = 0
4x – 5y + 7 = 0

Hence, 4x – 5y + 7 = 0 is the equation of the straight line that passes through (2, 3), has a slope of 4 / 5

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Conclusion

A point-slope form is a useful mathematical tool used to determine the equation of the straight line by using the point on the line and the slope of the line. In this article, we explain point-slope form with its mathematical expressions. We discussed the formula of the point-slope form of the line. Then we learn where the point-slope form originated.

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