What is the function of Slope and how it is applied on Point-Slope Form?

The slope is usually used for calculating the steepness of the line. It is also used to calculate the direction. In algebra, the slope is used to find the graph geometrically. The slope is a concept to find the direction of any line from left to right.

On the other hand, point-slope form is a way used to calculate the equation of a line. For calculating the equation of line point-slope form bis very preferable in this regard. For the calculation of the equation of the line by point-slope form, slope plays an important role.

What is the working of the Slope?

The slope is usually used in graphs and algebra. The slope is the sharpness of the line like the sharpness of plateaus or mountains or hills are said to be a slope. The slope is usually defined by the ratio of rise to run, rise is used for the vertical change like Δy = y 2 – y 1 , while the run is used for horizontal change like Δx = x 2 – x 1 .

The change in the two points of the vertical and horizontal points are representing the slope.
The slope is denoted by small m.

m = rise/run = Δy/ Δx = y 2 – y 1 / x 2 – x 1

Example 1

Evaluate the slope of the line of the given points, (18, -4) and (9, 14)?

Solution

Step 1: Identify the points of the line from the given problem.

x 1 = 18, x 2 = 9, y 1 = -4, y 2 = 14

Step 2: Write the general equation of the slope to calculate it.

m = (y 2 – y 1 ) / (x 2 – x 1 )

Step 3: Place the values in the above equation of the slope.

m = (14 – (-4)) / (9 – 18)

m = (14 + 4) / (-9)

m = 18/-9

m = -2

Example 2

Evaluate the slope of the line of the given points, (-42, -15) and (33, 35)?

Solution

Step 1: Identify the points of the line from the given problem.

x 1 = -42, x 2 = 33, y 1 = -15, y 2 = 35

Step 2: Write the general equation to calculate the slope.

m = (y 2 – y 1 ) / (x 2 – x 1 )

Step 3: Place the values in the above equation of the slope.

m = (35 – (-15)) / (33 – (-42))

m = (35 + 15) / (33 + 42)

m = 50/75

m = 10/15

m = 2/3

m = 0.6667

How slope is applied on Point-Slope form?

The slope is very essential in point-slope form. As by name of the point-slope form we can conclude that in the point-slope form we must have a slope and points. The point-slope form is usually a way to write the equation of the line accurately.

It is usually a linear form of the equation of the line. A linear equation of the form y – y 1 = m (x –x 1 ) is known as the point-slope form. In the point-slope form, the equation of the line can be calculated when a point of the line along with the slope of the line is given.

In the linear equation of the point-slope form, x and y are the fixed variables, x 1 and y 1 are the given points and m is the slope of the line. Now it is clear by the equation of point-slope form that, we can’t apply the point-slope form to calculate the equation of line without slope.
Slope can be in the form of integer, fraction, or decimal. To calculate the equation of the line with the help of point-slope form, we have to calculate the slope of the line by using the given points.

How to calculate the equation of the line by using the Point-slope form?

For the calculation of the point-slope form to determine the equation of the line, we must have points of the line, then calculate the slope of those points and then put the points along with the slope to the linear equation of the point-slope form.

To reduce the difficulty of such a large process of solving these kinds of problems. There is an online tool to calculate the accurate result of the given problem such as, the point slope equation calculator which gives you the accurate and perfect result of your problem.

Example 1

Find the point-slope form of the given points of the line, (-3, -9) and (7, -4)?

Solution

Step 1: Identify the points of the line from the given problem.

x 1 = -3, x 2 = 7, y 1 = -9, y 2 = -4

Step 2: Write the general equation to calculate the slope.

m = (y 2 – y 1 ) / (x 2 – x 1 )

Step 3: Place the values in the above equation of the slope.

m = (-4 – (-9)) / (7 – (-3))

m = (-4 + 9) / (7 + 3)

m = -5/10

m = -1/2

m = -0.5

Step 4: Now take the general equation of the point-slope form.

y – y 1 = m (x – x 1 )

Step 5: Put the value of the calculated slope and any pair of the points in the above linear

equation.

y – y 1 = m (x – x 1 )

y – (-4) = – ½ (x – 7)

y + 4 = -1/2 x + 7/2

y + ½ x + 4 – 7/2 = 0

y + ½ x + 8/2 – 7/2 = 0

y + ½ x + 1/2 = 0

y + 0.5x + 0.5 = 0

y = -0.5x – 0.5

Example 2

Find the point-slope form of the given points of the line, (9, -2) and (17, -4)?

Solution

Step 1: Identify the points of the line from the given problem.

x 1 = 9, x 2 = 17, y 1 = -2, y 2 = -4

Step 2: Write the general equation to calculate the slope.

m = (y 2 – y 1 ) / (x 2 – x 1 )

Step 3: Place the values in the above equation of the slope.

m = (-4 – (-2)) / (17 – (9))

m = (-4 + 2) / (17 – 9)

m = -2/8

m = -1/4

m = -0.25

Step 4: Now take the general equation of the point-slope form.

y – y 1 = m (x – x 1 )

Step 5: Put the value of the calculated slope and any pair of the points in the above linear equation.

y – y 1 = m (x – x 1 )

y – (9) = – 1/4 (x – (-2))

y – 9 = -1/4 (x + 2)

y – 9 = -1/4 x – 2/4

y + 1/4 x – 9 + 2/4 = 0

y + 1/4 x – 18/2 + 1/2 = 0

y + 1/4 x – 17/2 = 0

y + 0.25x – 8.5 = 0

y = -0.25x + 8.5

Summary

The slope is used to measure the sharpness of the line while the point-slope form is used to calculate the equation of the line. The slope is very helpful in calculating point-slope form. without slope, we are unable to calculate point-slope form.