The point-slope form is a method for expressing a linear line's equation. In this form, the equation of the line is specified by giving the coordinates of a single point on the line and its slope.

The origins of the point-slope form can be found in the work of French mathematician Augustin-Louis Cauchy, who made significant contributions to the study of functions, calculus, and geometry. Point slopes play an important role in the development of analytic geometry.

In this article, we will discuss the definition, derivation of point-slope form expression, steps to find the point-slope form, and solve related examples in detail.

What is the point slope form?

By using the slope and one point on the line, the equation of a straight line can be expressed in point-slope form. The point-slope form of a linear equation is a relatively modern concept in mathematics. 

The point-slope form can be used to determine a line's equation when additional information is provided in two parts. The first is a specific line point, while the second is either the slope or another line point.

Expression of Point Slope Form

The expression of a straight line's point-slope form is given as.

z – z1  = m (x – x1)

Where,

• (x1, z1) is a point lying on the line.
• “m” is called the slope of the line at a given point.
• (x, z) is the coordinates of any other point that also lies on the line.

In this form, the equation of the line is determined by a single point and its slope, which is useful to find the equation of a line by using a given point and slope. The point-slope form is used in mathematics, engineering, and science to represent a model relationship between variables that show linear growth or decay.

Point-slope form expression's interpretation

The point-slope form expression of a linear equation can be derived using the two points-slope formula of the linear line.

The formula for the linear line's two points-slope is as follows: 

m = (z2 – z1)/(x2 – x1)

Where (x1, z1) and (x2, z2) are two points located on the line.

Let (x1, z1) & (x, z) be two known points on the line, and let "m" be the slope of the line to obtain the point-slope form expression. The two points slope formula then becomes. 

m = (z – z1)/(x – z1)

By Cross-multiplying we get.

z – z1 = m (x - x1)

Which is the point-slope form of the linear equation.

How to locate a point-slope form?

These steps can be used to determine a linear equation's point-slope form.

• Locate the line's well-known point. Let (x1, z1) represent the location of this point.
• Locate the slope of the line. If the slope is given, then simply use that value. 
• If the slope is not supplied, locate another point on the line to determine the slope of the line, that point is in the form (x2, z2), and use the two points-slope formula to find the slope.

m = (z2 – z1)/(x2 – x1)

• Substitute the values of the known point (x1, z1) and slope of line into the point-slope formula.

z – z1 = m(x – x1)

• If the point (x1, z1) is not given then it is found using the formula of slope-intercept form using its slope and z-intercept. The formula for the slope intercept is given below. 

z = mx + b, where “m” is the slope of the line and “b” is the z-intercept.

• Find the point (x1, z1) by the above formula and put it into the point-slope formula to find the desired result. 

By following the above steps, we find the equation of a straight line in point-slope form by putting the given point (x1, z1), and the slope “m” of the line.

How to calculate point slope form?

In this section, we discuss the different examples with detailed steps to find the point-slope form of the line.

Example 1:

Derive the point-slope form expression of the line whose slope is “3” and the point is (1, 3).

Solution:

Step 1: write the given data carefully.

Slope of line = m = 3, given point = (x1, z1) = (1, 3)

Noted, x1 = 1 & z1 = 3.

Step 2: The point-slope form's formula is written as.

z – z1  = m (x – x1)

 Step 3: Put the values carefully in the above formula.

x1 = 1, z1 = 3 & m = 3.

z – 3 = 3 (x – 1)

z – 3 = 3 (x – 1) that is the required point-slope form of the line.

 Example 2: 

Determine the expression of a point-slope form of the line at points (-1, 4) and (2, -3).

Solution:

Step 1: write the given data carefully.

 (x1 , z1) = (-1, 4), (x2, z2) = (2, -3)

Noted, x1= -1, x2 = 2, z1 = 4, z2 = -3 

Step 2: Write the two-point slope formula.

m = (z2 – z1)/(x2 – x1)

 Step 3: Put the value from step 1 in the above formula and simplify.

x1= -1, x2 = 2, z1 = 4, z2 = -3

m = [(-3)– (4)]/[(2) – (-1)]

m = [-3– 4]/[2 +1]

m = [-7]/[3]

m = -7/3

Step 4: Write the point-slope formula.

z – z1  = m (x – x1)

Step 5: Put the value of slope in the above formula at a point (-1, 4) carefully and simplify.

m = -7/3, x1= -1, z1 = 4.

z – (4)= (-7/3)[x – (-1)]

z - 4 = -7/3 [x + 1]

z - 4 = -7/3 [x + 1] is the point-slope form of the line that passes through points (-1, 4) and

 (2, -3)

You can also take help from a point slope equation calculator to evaluate the problems of point slope form to determine line equation.

Example solved through point slope form calculator by Allmath

Summary

In this article, we discussed the definition and the derivation of the point-slope form formula. Moreover discussed the steps to find the slope-point form in a different situation. For a better understanding of the concept of point-slope form solved different examples in detail.