Slope vs Slope Intercept Form vs Point Slope Form

Posted On October 20 2021

Slope intercept is an algebra topic. Linear equations of straight lines used slope-intercept form. It is a pretty simple topic in which we have to find slope and y-intercept and draw a graph. Everyone easily finds the slope and y-intercept.

Table of Contents

What is Slope?

The slope is the measure of the steepness of the line. A slope of the line is a number that describes both the direction and steepness of a line. It is also known as a gradient.

Slope is denoted by a letter m.

Slope is calculated by finding the ratio of vertical change to the horizontal change between any two distinct points on a line. This online slope calculator can ease up your slope calculations and draw a graph with the given input.

$Slope = \frac{Rise}{Run}= \frac{\Delta y}{\Delta x}= \frac{y_2-y_1}{x_2-x_1}$

Example:

Find the slope of the points (11, 14), (12, 15)?

Solution:

We know that (), ()

From the given points we have:

$^{x_1}= 11$

$^{x_2}= 12$

$_{y_1}= 14$

$^{y_2}= 15$

We have the formula of finding the slope

$m = \tfrac{y_2-y_1}{x_2-x_1}$

by putting the values, we get

$m= \frac{15-14}{12-11}$

$= \frac{1}{1}= 1$

Hence the slope m = 1

Example:

Find the slope of the points (21, -14), (12, -15)?

Solution:

We know that point (), ()

From the given points we have:

$^{x_1}=21$

$^{x_2}=12$

$_{y_1}=-14$

$^{y_2}= -15$

We have the formula of finding the slope:

$m = \tfrac{y_2-y_1}{x_2-x_1}$

by putting the values, we get

$m = \tfrac{-15-(-14)}{12-21}$

$= \frac{-15+14}{12-21}$

$= \frac{-1}{-9}$

$= \frac{1}{9}$

Hence the slope $m= \tfrac{1}{9}$

What is the Slope Intercept form?

It is the most frequent way to represent the equation of a line.

The equation of a straight line in the form of

y = mx + b

Where m is the slope of line and b is its y-intercept is known as slope-intercept form.

If you know the slope m and y-intercept (0, b) of a line you can write the equation of a line in slope-intercept form.

Example:

What is the equation of the line in slope-intercept form?

y = -5x + 3

Solution:

From the given equation we have

m = -5 (slope)

b = 3 (y-intercept)

The line is decreasing from left to right due to a negative slope.

And passing at point (0, 3) through the y-axis.

You can refer to the slope intercept form calculator for step-by-step calculation.

Example:

What is the slope-intercept form of a line passing through the points (21, -14) and (12, -15)?

Solution:

We know that points (), ()

From the given points we have

$^{x_1}=21$

$^{x_2}=12$

$_{y_1}=-14$

$^{y_2}= -15$

We have the formula of finding the slope

$m = \tfrac{y_2-y_1}{x_2-x_1}$

by putting the values, we get

$m= \tfrac{-15-(-14)}{12-21}$

$= \frac{-15+14}{12-21}$

$= \frac{-1}{-9}$

$= \frac{1}{9}$

Hence the slope $m= \tfrac{1}{9}$

Put the value of m in the equation of the slope-intercept form

$^{y}= mx+b$

$y=\tfrac{1}{9}+b$

Now put any point to this equation

Let us put the points (12, -15)

$-15 = 1/( 9)(12)+ b$

$-15 - 1/( 9)(12) = b$

$-15 - 4/( 3) = b$

$- 49/( 3) = b$

Put the value of b in slope equation

$y = 1/( 9) x+ - 49/( 3)$

What is Point slope form?

The equation of a straight line in the form of

y – y₁ = m (x – x₁)

Where m is the slope of line and (x₁, y₁) are the coordinates of a given point on the line. This equation is known as point slope form

Example:

Find the point slope form of the point (21, -14) at m = 1?

Solution:

We know that point ()

From given points we have

$x_1=21$

$y_1=-14$

We have the formula of finding the point-slope form:

$y_ -y_1=m (x_ -x_1)$

By putting the values, we get

$y_ -(-14)=1 (x_ -21)$

$y_ +14= x_ -21$

$y_ = x_ -21-14$

$y_ = x_ -35$

Here m = 1 and b = -35

Example:

Find the point slope form of the point (12, 15) at m=1/9?

Solution:

We know that point ()

From given points we have

$x_1=12$

$y_1=15$

We have the formula of finding the point slope form

$y_ -y_1=m (x_ -x_1)$

By putting the values, we get

$y_ -15=1/9 (x_ -12)$

$y_ -15=1/9 x_ -12/9$

$y =\frac{1}{9x}x-\frac{4}{3}+15$

$y =\frac{1}{9} x -\frac{41}{3}$

Here m = 1/9 and b = -41/3

Two-point slope form:

The equation of straight line in the form of

$y -y_1=\tfrac{y2-y1}{x2-x1}(x_ -x_1)$

$y -y_2=\frac{y2-y1}{x2-x1}(x_ -x_2)$

These equations are known as two-point slope form.

Keep (x, y) as variable and (), () are points

This equation also be written as

$\left (\frac{y-y1}{y2-y1}= \frac{x-x1}{x2-x1} \right )$

Example:

Find the point slope form of the point (21, -14), (12, -15)?

Solution:

We know that point (), ()

From given points we have

$x_1=21$

$x_2=12$

$y_1=-14$

$y_2=-15$

We have the formula of finding the point-slope form:

$y_ -y_2=\frac{y2-y1}{x2-x1}(x -x_2)$

By putting the values, we get:

$y -15=\frac{-15-(-14))}{12-21}(x -12)$

$y -15=\frac{-15+14}{12-21}(x -12)$

$y -15=\frac{-1}{-9}(x -12)$

$y -15=\frac{1}{9}(x -12)$

$y -15=\frac{1}{9}x-\frac{12}{9}$

$y =\frac{1}{9}x-\frac{4}{3}+15$

$y=\frac{1}{9}x-\frac{41}{3}$

Here m = 1/9 and b = – 41/3.

Blogs -> Slope vs Slope Intercept Form vs Point Slope Form

Slope vs Slope Intercept Form vs Point Slope Form

What is Slope?

What is the Slope Intercept form?

What is Point slope form?

Two-point slope form:

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