## Table of Contents

- What is Slope?
- The Slope Formula and Rise Over Run
- Figuring Out Types of Slopes
- How to Find Slope With Two Points
- Slope-Intercept Form
- Lesson Summary

Learn how to determine the slope of a line with two points, how the slope formula is related to rise over run, and review examples of how to find the slope.
Updated: 07/18/2021

- What is Slope?
- The Slope Formula and Rise Over Run
- Figuring Out Types of Slopes
- How to Find Slope With Two Points
- Slope-Intercept Form
- Lesson Summary

The **slope**, also referred to as the gradient in mathematics, is a value given to the steepness of a line. Within two connecting points, the slope measures the steepness and direction of the line. Denoted as **m**, the value of the slope is directly proportional to the steepness. In other words, as the value of the slope increases, the steepness increases.

To find the slope of a line with two points, the coordinates of any two points passing through the line must be identified. Using these coordinates, the amount of change of the *y*-coordinates must be divided by the amount of change of *x*-coordinates, resulting in the slope or ** m**. The value of a slope is not only taken into consideration in mathematics, but it can also be applied in real life. For instance, slopes can be applied by civil engineers while building roads.

The slope of a line may also be defined as rise over run or vertical change over horizontal change. As shown in the figure below, in relation to the slope formula, the rise is associated with the change of the *y*-coordinates while the run is associated with the change of the *x*-coordinates. The slope can be found by identifying the **rise**, or vertical change, and dividing it by the obtained **run**, or the horizontal change.

The slope formula can be used to obtain the slope of a line through the following steps:

- Identify any two points on the line.
- According to the slope formula, denote the first point as (x1, y1)
- Denote the second point as (x2, y2).
- Substitute the newly denoted coordinates into the formula to obtain the slope or
**m**.

In relation to the rise over run formula, the change of the *y*-coordinates can be defined as the rise while the change of the *x*-coordinates can be defined as the run. The two approaches involve dividing the vertical factors over the horizontal factors to obtain the slope. Additionally, both approaches allow the usage of any two points within the given line to find the slope.

Depending on the two points used to make a line, the slope can have several types:

A positive slope is represented by an increasing line that is rising upwards from left to right.

Example: A line with the two following points (2,2) and ( -1, -2) has a positive slope of {eq}\frac{4}{3}{/eq}

A negative slope is represented by a decreasing line that is declining downwards from left to right.

Example: A line with the two following points (0,6) and (3, 3) has a negative slope of -1.

A slope of zero is represented by a horizontal and constant line.

Example: A line with the two following points (-1,5) and (1, 5) has a slope of zero.

An unidentified slope is represented by a vertical line.

Example: A line with the two following points (2,3) and (2, 1) has an undefined slope.

The slope of a line can be determined using the slope or rise over run formula using any two points identified on that line. Let's look at a couple of examples.

Example 1: Suppose a line contains the two following points (8,4) and (-3,7). What is the slope of the line?

- Identify the two points on the line as (8,4) and (-3,7).
- According to the slope formula, denote the first point (8,4) as (x1, y1).
- Denote the second point (-3,7) as (x2, y2).
- Substitute the newly denoted coordinates into the formula to obtain the slope or
**m**.

- Final answer: m = {eq}\frac{4-7}{8-(-3)}{/eq} = {eq}\frac{-3}{11}{/eq}

Example 2: Suppose a line contains the two following points (1,9) and (2,12). What is the slope of the line?

- Identify the two points on the line as (1,9) and (2,12).
- According to the slope formula, denote the first point (1,9) as (x1, y1).
- Denote the second point (2,12) as (x2, y2).
- Substitute the newly denoted coordinates into the formula to obtain the slope or
**m**.

- Final answer: m ={eq}\frac{12-9}{2-1}{/eq} = 3

Any two points on a line can be presented on the Cartesian coordinate system to represent the slope of a line. According to the Cartesian coordinate system, the two points (x1, y1) and (x2, y2) are plotted to form a right triangle with a given angle of inclination. Two sides of the triangle can be identified using the change in *x* and *y* values. However, the third side of the triangle is also known as the side **d** and formed by the distance between (x1, y1) and (x2, y2). The value of **d** can be calculated through the following Pythagorean theorem equation:

Through the Cartesian coordinate system, the points within a line may be distributed upon the four different quadrants according to the type of slope.

- Example 1: According to the Cartesian coordinate system, a line with a positive slope of two points (0,1) and (13,9) will be passing through the first and third quadrants.
- Example 2: According to the Cartesian coordinate system, a line with a negative slope of two points (0,4) and (2,1) will be passing through the second and fourth quadrants.
- Example 3: According to the Cartesian coordinate system, a line with a slope of zero of two points (1,2) and (-1,2) will be passing through the first and second quadrants.
- Example 4: According to the Cartesian coordinate system, a line with an undefined slope of two points (-1,3) and (-1, -1) will be passing through the second and third quadrants.

The equation of a line may be represented in different forms, including the slope-intercept form, which is given by *y* = *mx* + *b*. The term *m* in the equation presents the slope, while the term *b* presents the *y*-intercept.

Example 1: find the slope in the line y = 4x + 5.

The slope would be 4. This is because in the slope-intercept form (y = mx + b), 4 is in the place of 'm'.

Example 2: find the slope in the line 3y = 9x + 15.

In order to get this equation in its slope-intercept form, every component must be divided by 3 to yield the line y = 3x + 5. Thus, the slope is 3.

Example 3: find the slope in the line 6x = 5 -y.

Again, to get the equation in its slope-intercept form, "y" must be solved for. After solving for "y", the equation reads y = -6x +5. By using the slope-intercept form, the slope is found to be -6.

The **slope** is a mathematical value identifying the steepness and direction of a given line.

- Various associated equations with similar definitions can be used to calculate the value of the slope, including the slope of a line of two points.
- Within calculating the value of the slope either through the slope formula or the rise over run form, the resulting slopes may be within the four types (positive, negative, zero, and undefined).
- Any two given points on a line can also be applied to the Cartesian coordinate system along with the Pythagorean theorem to find the distance between the two points.
- Finding the value of the slope can be used to construct the equation of a line in different forms including the slope intercept form.

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Additional Activities

Find an equation of the line through the point (5,2) parallel to the line 4x +6y +5 = 0

Show that the lines 2x + 3y = 1 and 6x -4y -1 = 0 are perpendicular.

a) Express temperature T as a function of height h, assuming a linear relationship. At ground level, T=20 degrees Celsius. At h = 1 km., T = 10 degrees Celsius.

b) What is the temperature at 2.5 km ?

Rewrite the equation as y = -2x/3 -5/6.

This equation is now in the form of the slope intercept formula ( y = mx + b ). m = -2/3.

Parallel lines have the same slope. The equation in point-slope form is

y-2= (-2/3)(x-5). We can rewrite the equation as

2x + 3y = 16.

Rewrite these equations in slope-intercept form:

y= -2x/3 + 1/3 and y = 3x/2 - 1/4

The slopes are -2/3 and 3/2.

The product of the two slopes is -1,

which is what is required for two lines to be perpendicular.

a) T = mh + b is the slope intercept form.

20 = m(0) + b; The intercept b is 20.

10 = m(1) + 20; The slope is -10. The equation is T = -10 h + 20.

b)

At h = 2.5 km, T = -10(2.5) + 20 = -5 degrees Celsius.

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