In algebra, there are several ways to write, express, or find the linear equation of a straight line. A linear equation is an equation in which the power of variables is always 1. The different ways for expressing the linear equation are slope intercept form, point-slope form, and x & y-intercept form.

Algebra students mostly preferred slope intercept form to express the linear equation by taking the slope and y-intercept of the line. In this post, we are going to express the term point slope form with its definition, formula, and how to determine the linear equation along with a lot of examples.

What is the point-slope form?

In algebra, a line that is expressed or determined with the help of two coordinate points of the line or slope & one coordinate point line of the line is said to be the point-slope form. It is widely used to determine the straight-line equation.

The term slope and coordinate points are to be taken from the given line. The slope of a line is the measure of the steepness of the line with the help of coordinate points. It is evaluated by placing the x & y coordinate points in the below formula.

Slope = m = (y2 – y1)/(x2 – x1)

Use the above formula to evaluate the slope of a line for point slope form with the help of coordinate points of a line if it is not given. Or you can also find the slope of a line online with the help of a slope calculator.

Formula of the point-slope form

The general expression of the point-slope form is:

y – y1 = m (x – x1)

where

• m = slope of the line
• x & y = fixed points of the line
• x1 & y1 = coordinate point of the line

The general expression of point slope form is derived from the formula of a slope by multiplying both sides by (x2 – x1).

m * (x2 – x1) = (y2 – y1)/(x2 – x1) * (x2 – x1)

m * (x2 – x1) = (y2 – y1)

How to determine a line equation through point-slope form?

The linear equation of the line can be evaluated with the help of a general expression of the point-slope form by placing the values of the slope of a line and 1 coordinate point of a line. Let us take a few examples of determining line equations through two points and slope & 1 point.

For 1 point and slope

Example

Express the straight line equation with the help of point slope form by using the 1 & slope method if the slope of a line is 4 and the coordinate point is (3, 6).

Solution

Step 1: Firstly, take the given values of the slope and coordinate points of a line.

Slope = m = 4

x1 = 3

y1 = 6

Step 2: Now take the general expression of the point-slope form.

y – y1 = m (x – x1)

Step 3: Substitute the given point and the slope of a line to the above expression to determine the linear equation of a line through point slope form. y – y1 = m (x – x1)

y – 6 = 4 * (x – 3)

y – 6 = 4 * x – 4 * 3

y – 6 = 4x – 12

y – 6 - 4x + 12 = 0

y – 4x + 6 = 0

4x – y – 6 = 0

Cross-check the result of the above problem by using a point slope form calculator by Allmath.

For two points

Example 1

Express the straight line equation with the help of point slope form by using x & y coordinate points: (14, 18) & (24, 36)

Solution

Step 1: Firstly, take the given values of the x & y coordinate points of a line.

x1 = 14, x2 = 24, y1 = 18, y2 = 36

Step 2: As the slope of the line is not given, calculate it first by taking the general formula of the slope.

Slope of the line = m = y2 – y1 / x2 – x1

Now substitute the points of the x & y coordinate of the line to the above formula.

Slope = m = (36 – 18) / (24 – 14)

Slope = m = (18) / (10)

Slope = m = 18 / 10

Slope = m = 9/5

Slope = m = 1.8

Step 3: Now take the general expression of the point-slope form.

y – y1 = m (x – x1)

Step 4: Substitute the given point and the calculated slope of a line to the above expression to determine the linear equation of a line through point slope form. y – y1 = m (x – x1)

y – (18) = 1.8 * (x – 14)

y – 18 = 1.8x – 1.8 * 14

y – 18 = 1.8x – 25.2

y – 18 – 1.8x + 25.2 = 0

y – 1.8x + 7.2 = 0

1.8x – y – 7.2 = 0

Example 2

Express the straight line equation with the help of point slope form by using x & y coordinate points: (-4, -8) & (12, 16)

Solution

Step 1: Firstly, take the given values of the x & y coordinate points of a line.

x1 = -4, x2 = 12, y1 = -8, y2 = 16

Step 2: As the slope of the line is not given, calculate it first by taking the general formula of the slope.

Slope of the line = m = y2 – y1 / x2 – x1

Now substitute the points of the x & y coordinate of the line to the above formula.

Slope = m = (16 – (-8)) / (12 – (-4))

Slope = m = (16 + 8) / (12 + 4)

Slope = m = (24) / (16)

Slope = m = 12 / 8

Slope = m = 3/2

Slope = m = 1.5

Step 3: Now take the general expression of the point-slope form.

y – y1 = m (x – x1)

Step 4: Substitute the given point and the calculated slope of a line to the above expression to determine the linear equation of a line through point slope form. y – y1 = m (x – x1)

y – (-8) = 1.5 * (x – (-4))

y + 8 = 1.5 * (x + 4)

y + 8 = 1.5x + 1.5 * 4

y + 8 = 1.5x + 6

y + 8 – 1.5x – 6 = 0

y – 1.5x + 2 = 0

1.5x – y – 2 = 0

Conclusion

Now you can understand the point-slope form easily as we have discussed the definition, formula, derivation, and solved examples. Now you just need to take an overview of this post and practice the problems by hand for once to master it.