Mathematical Plane

Before we dive into what a slope is, it’s important to understand what a plane is in mathematics. When we talk about planes, we are usually referring to a Cartesian plane. Check one out below.

 

axes_quadrants_examples

There are a couple of characteristics that all graphs have. Let’s take a look at some of those characteristics below.

 

Characteristic Description
1 Four quadrants. These quadrants go counter-clockwise.
2 Two axes: an x axis and a y axis.
3 There are only four directions you can go in: up or down, left or right.

 

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Distance Between Two Points Formula

It is important to know the formula you use to find the distance between two points. First, you should understand what points are on a graph.

 

Formal Name Definition Standard Form Explanation Example
Point Coordinates Coordinates that give you directions to how to get to a point on a graph (x,y) (x-coordinate, y-coordinate) (-5, -2)

 

As you can see from the definition above, you can have almost any combination of points as long as they give you at least an x-coordinate and a y-coordinate.

 

points_axes

 

x-coordinate y-coordinate
A -4 2
B 2 2

 

To find the distance between these two points, you simply have to follow the distance formula.

 

point_distance_formula

 

x_{1} y_{1} x_{2} y_{2}
-4 2 2 2

 

Equation of a Line

Now, let’s introduce the concept of a slope. The term slope in maths is similar to what you encounter in real life: like the slope of a hill. The only difference is that in maths we’re referring to lines. Let’s take a look at the standard equation of a line.

 

slope_intercept_formula

 

This is called the slope-intercept formula because it includes the y-intercept (b) and a slope.

 

The point (x,y)
Slope m
Y-intercept b

 

Slope Formula

The slope can be thought of as the incline or decline of the line. You can think of the slope of a line as the amount you rise over the amount you run.

 

rise_over_run_explanation

 

In reality, we can go in two directions.

 

Positive Direction Negative Direction
y-axis Rise (Up) Down
x-axis Run (Right) Left

 

Slope Formula Given Two Points

To have a slope, you need to have at least two points. This is because any two points form a line. If you just have one point, you can’t really have a slope because there are infinitely many directions you can go from that point.

 

To find the slope between two points, you should follow the slope formula.

 

m (x_{1}, y_{1}) (x_{2}, y_{2}) m = frac{y_{2}-y_{1}}{x_{2}-x_{1}}
slope Point 1 Point 2 Slope formula

 

Slope Formula of Parallel Lines

Parallel lines are one of the two special types of lines. In order to have parallel lines, you have to satisfy the following conditions.

 

parallel_lines_example

 

A Condition 1 Lines are equal distance to each other always
B Condition 2 Lines don’t ever touch, no matter how long they go for

 

Finding the slope between two parallel lines is the easiest to find. Check out the rules below.

 

Two lines are parallel The slopes of the two lines are equal y=(m)x+b

y=(j)x+b

m=j

 

Slope Formula of Perpendicular Lines

Perpendicular lines are the second type of line. When two lines touch, they are said to intersect. This is easy enough to remember - just think of a street intersection. A street intersection is where two streets meet each other.

 

Condition 1 Perpendicular lines are intersecting
Condition 2 They have a point of intersection
Condition 3 Their lines form a right angle (90 degrees)

 

Perpendicular lines are a special type of intersecting lines. Finding the slope between two perpendicular lines is a bit harder.

 

Two lines are perpendicular The slopes of the two lines are reciprocals y=(m)x+b

y=(j)x+b

(j)=reciprocal of (m)

 

In order to find the reciprocal, you should take a look at the table below.

 

Definition
Reciprocal When a number (m) and it’s reciprocal (j) are multiplied together, they equal 1.
Reciprocal of positive number Reciprocal of m = frac{1}{m}

m*frac{1}{m} = 1

Reciprocal of negative number Reciprocal of -m = - frac{1}{m}

-m*- frac{1}{m} = 1

 

Example 1

Let’s go through a step-by-step example of how to find the slope of the following two points.

 

Point 1 (3, 6)
Point 2 (-2, 8)

 

In order to find the slope of two points, we must use the formula of the slope between two points. Recall that this is simply the two y coordinates subtracted from each other divided by the two x coordinates subtracted by one another.

 

[

m = frac{y_{2}-y_{1}}{x_{2}-x_{1}} =  frac{8-6}{-2-3} = frac{2}{-5}

]

 

The slope between these two points is frac{2}{-5}.

 

Example 2

Using what you know about parallel lines, give an equation of a line that is parallel to the following line.

 

[

y=-3x+10

]

 

In order to solve this, we simply need to remember that parallel lines have the same slope. Therefore, we can make any combination of lines.

 

1 y=-3x
2 y=-3x+1
3 y=-3x-15

 

Example 3

Find the slope of any line perpendicular to the following line.

 

[

y= -frac{1}{3}x + 1

]

 

We simply need to find the reciprocal.

 

The slope Divide 1 by the slope Check
-frac{1}{3} 1 div -frac{1}{3} = -3 -frac{1}{3}*-3 = 1

 

Any line we make with a slope of -3 will be perpendicular to this line.

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.