Slope vs Slope Intercept Form vs Point Slope Form

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.

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.