Two Variable Linear Equations: Slope of a Line

I. Intro: The Slope Of A Line

The mathematical definition of slope is very similar to our everyday one. In math, slope is used to describe the steepness and direction of lines. By just looking at the graph of a line, you can learn some things about its slope, especially relative to other lines graphed on the same coordinate plane. Consider the graphs of the three lines shown below:

First, let’s look at lines A and B. If you imagined these lines to be hills, you would say that line B is steeper than line A. Line B has a greater slope than line A.

Next, notice that lines A and B slant up as you move from left to right. We say these two lines have a positive slope. Line C slants down from left to right. Line C has a negative slope. Using two of the points on the line, you can find the slope of the line by finding the rise and the run.

II. Finding Slope From a Graph
As we saw in the previous section, the rate of change determines whether a set of data is linear or non-linear. In this section, we formalize how to find this rate of change.

The rate of change of or the distance between the y-values compared to the x-values is called the slope. The symbol for slope is 𝑚. As we will see, graphically, the slope measures how steep the line is.

One way to determine the slope is to plot the points on a graph, draw a line through them, and then count the change in the y-values, or rise, and the change in the x-values, or run.

Rise - Vertical change between two points. 

Run - Horizontal change between two points. 

The steps to finding the slope from a graph are:

1) Select any two random points on the graph of the line (preferably with integer coordinates).
2) Label them as A and B (in any order).

3) Calculate the "rise" from A to B. While going vertically from A to B, if we have to go
"up", then the rise is positive;
"down", then the rise is negative.
4) Calculate the "run" from A to B. While going horizontally from A to B, if we have to go
"right", then the run is positive;
"left", then the run is negative.
5) Now, use the formula: slope = rise/run

Example 1

Example 2

Example 3

Khan: Intro to Slope

III. Finding The Slope From Two Points

You can find the slope of the line by counting the rise over run. As we've alluded to above, you can also find the slope of a straight line without its graph if you know the coordinates of any two points on that line. Every point has a set of coordinates: an x-value and a y-value, written as an ordered pair (x, y). The x value tells you where a point is horizontally. The y value tells you where the point is vertically.

Consider two points on a line—Point 1 and Point 2. Point 1 has coordinates (x₁,y₁) and Point 2 has coordinates (x₂,y₂).

The rise is the vertical distance between the two points, which is the difference between their y-coordinates. That makes the rise (y₂−y₁). The run between these two points is the difference in the x-coordinates, or (x₂−x₁).

An alternate way to present the formula is:

Slope = Δy/Δx

Example

In the example you’ll see that the line has two points each indicated as an ordered pair. The point (0,2) is indicated as Point 1, and (−2,6) as Point 2. So you are going to move from Point 1 to Point 2. A triangle is drawn in above the line to help illustrate the rise and run.

You can see from the graph that the rise going from Point 1 to Point 2 is 4, because you are moving 4 units in a positive direction (up). The run is −2−2, because you are then moving in a negative direction (left) 2 units. Using the slope formula,

Slope = rise/run = 4/-2 = -2

You do not need the graph to find the slope. You can just use the coordinates, keeping careful track of which is Point 1 and which is Point 2. Let’s organize the information about the two points:

Can Either Point Be (X₁, Y₁)?

Let's find out.

IV: Positive & Negative Slopes

When we look at a line on a graph, its slope can fall into one of four categories. Each type of slope tells us something different about how the line behaves:

1. Positive Slope: A line with a positive slope rises as it moves to the right on a coordinate plane. It shows that as the x-values increase, the y-values also increase. Think of it as going uphill.

2. Negative Slope: A line with a negative slope falls as it moves to the right on a coordinate plane. It shows that as the x-values increase, the y-values decrease. Think of it as going downhill.

3. Zero Slope: A flat, horizontal line has a zero slope. It means there’s no change in y-values as the x-values increase. The line stays level.

4. Undefined Slope: A vertical line has an undefined slope. This is because the x-values don’t change, while the y-values might change infinitely. 

Khan: Positive & Negative Slope

CK-12: Introduction: Linear Equations and Inequalities of Two Variables

CK-12: Introduction to Linear Equations In Two Variables

Lumen: Slope of a Line

Cuemath: How To Find Slope From A Graph

Mathnasium: What is a Negative Slope

https://www.mathwarehouse.com/algebra/linear_equation/slope-of-a-line.php

Algebra Supporting Notes

When I started relearning pre-algebra, I used the pre-algebra section in Khan Academy as a guide. I went through the videos and found other written sources along the way. The problem with this approach is that I find Khan Academy's presentation lacking clarity. For that reason, I'm using this section to sort of reformat things by starting my conceptual understanding of the subject matter from scratch. 

I. What is Algebra?

Algebra is a branch of mathematics that deals with symbols (variables) and the rules for manipulating those symbols to represent and solve problems involving relationships between quantities. These symbols typically represent numbers, and they allow us to write mathematical expressions and equations in a generalized form. Key features include use of variables, expressions, equations, inequalities, etc. 

II. What are the different types of equations in beginning algebra?

1. Linear Equations

• Form: ax + b = 0

• Description: These equations involve variables with a power of 1 (linear) and are graphically represented as straight lines.

2. Quadratic Equations

• Form: ax² + bx + c = 0

• Description: These equations involve a variable squared ($x^2$) and are graphically represented as parabolas.

3. Absolute Value Equations

• Form: $\mathrm{\mid }ax+b\mathrm{\mid }= c$

• Description: These equations involve the absolute value of an expression, resulting in two potential solutions.

4. Rational Equations

• Form: $\frac{P(x)/}{Q(x) }= R(x)$
  

• Description: These equations involve fractions with polynomials in the numerator and/or denominator. Solutions must exclude values that make the denominator zero.

5. Radical Equations

• Form: √x + a = b

• Description: These equations include variables within a square root (or other roots). Solutions may need to be checked for extraneous results.

6. Exponential Equations

• Form: a² = b

• Description: These equations have variables as exponents. They often require logarithms to solve.

7. Inequalities (Optional in Some Courses)

• Form: $a\mathrm{x }+ \mathrm{b }> c$

• Description: These are similar to equations but involve inequality signs ($>, <, \ge , \le$).

8. Proportions

• Form: $\frac{\mathrm{a/}}{\mathrm{b }}= \frac{\mathrm{c/}}{d}$

• Description: These involve two ratios set equal to each other and are solved by cross-multiplication.

9. Systems of Equations

• Form:

  • Linear system:

    
    
    
    
    

• Description: These involve solving two or more equations simultaneously.

Understanding these types helps in identifying the approach and techniques needed to solve different algebraic problems.

III. Are linear expressions also polynomials?
Yes

A polynomial is any expression made up of variables and constants using only:

• Addition

• Subtraction

• Multiplication

• Non-negative integer exponents

A linear expression has the form:

ax+b

Where:

a and b are constants

x is the variable

The exponent on x is 1 (which is allowed in polynomials)

All linear expressions are polynomials,
but not all polynomials are linear expressions.

Two Variable Linear Equations

I. Intro

A two variables equation is any equation that includes two variables (e.g., x and y). In this section we will focus on two variable linear equations.

In linear equations with one variable the expression is compared to a fixed quantity whereas in linear equations with two variables, a relationship exists between two variables, a change in one is often followed by a change in the other.

A linear equation with two variables can be written in the form ax + by + c = 0, where a, b, c ∈ ℝ, a ≠ 0 and b ≠ 0 and x and y are variables, each raised to the first power only.
 

Here are some examples of both linear and non-linear equations.

Solutions to equations of one variable are values that make the equation true. Similarly, solutions to equations with two variables are pairs of 𝑥- and 𝑦-values that make the equation true. There are an infinite number of possible combinations of 𝑥- and 𝑦-values that will satisfy the equation.

II. Graphing a Linear Equation

The graph of a linear equation in one variable x forms a vertical line that is parallel to the y-axis and vice-versa, whereas, the graph of a linear equation in two variables x and y forms a straight line. Let us graph a linear equation in two variables with the help of the following example.

Example: Plot a graph for a linear equation in two variables, x - 2y = 2.

Let us plot the linear equation graph using the following steps.

Step 1: Convert the equation in the form of y = mx + b. This will give: y = x/2 - 1.

Step 2: Now, we can replace the value of x for different numbers and get the resulting value of y to create the coordinates. When we put x = 0 in the equation, we get y = 0/2 - 1, which results in. y = -1. Similarly, if we substitute the value of x as 2 in the equation, y = x/2 - 1, we get y = 0. If we substitute the value of x as 4, we get y = 1. The value of x = -2 gives the value of y = -2.  

Step 3: Create a table with the various coordinates.

X  |  Y

-2  |  -2

0   |  -1

2   |   0

4   |  -2

Step 4: Finally, we plot these points on a graph and draw a line through them.

III. Intercepts

The intercepts of a line are the points where the line intercepts, or crosses, the horizontal and vertical axes. To help you remember what “intercept” means, think about the word “intersect.” The two words sound alike and in this case mean the same thing.

The straight line on the graph below intercepts the two coordinate axes. The point where the line crosses the x-axis is called the x-intercept. The y-intercept is the point where the line crosses the y-axis.

x intercept: x value when y = 0

y intercept: y value when x = 0

A. Intercepts From A Graph

Finding the x and y intercepts from a graph is simply a matter of finding where the line crosses the horizontal and vertical axis. 

The x intercept is the point where the line crosses the horizontal x axis at (5, 0).

The y intercept is the point where the line crosses the vertical y axis at (0, 4).

B. Intercepts From An Equation
Notice that the y-intercept always occurs where x = 0, and the x-intercept always occurs where y = 0.

To find the x and y intercepts of a linear equation, you can substitute 0 for y and for x respectively.

For example, 3y + 2x = 6 

Solve for y = 0

3(0) + 2x = 6

2x = 6

x = 3

So the x-intercept is (3,0).

Then solve for x = 0.

3y+2(0)=6

3y = 6

y=2

The y-intercept is (0,2).

Now that you have the coordinates (3, 0) and (0, 2) you can use them to graph the line.

IV. Recognize Linear Equations

We can recognize linear equations by how the input or independent variable (𝑥 variable), and output or dependent variable (𝑦 variable) values change with respect to each other. By change, we mean what is the distance between known values. The quantity we use to compare changes is known as the rate of change. A rate is a type of ratio or fraction in which you are comparing two quantities often with different units. We see a lot of these in our daily life (60 mph, $3.99 per lb.).

Rate of Change = difference in dependent variable values/difference in independent variable values

In the following sections, rate of change will be called slope and we will find the ratio of the change in y-values to the change in x-values. 

For linear models, the rate of change is constant, that is, does not change, regardless of which data points you choose to consider. Linear relationships have a constant rate of change. If the rate of change is not constant, the relationship is not linear.

For the moment, we will consider data without units.

Example 1
Determine whether the relation in the table below is linear or non-linear.

X  |  Y
0   |  3

1   |  6

2   |  9

3   |  12

Solution: The x-values are going up by one unit at a time, while the y-values are going up by 3 units at a time. Therefore, the slope is: 3/1 = 3. Since there is a constant rate of change, the relation in the table is linear.

Example 2

Determine whether the relation in the table below is linear or non-linear.

X  |  Y

-1 |  10

0  |  7

1  |  4

2  |  1

Solution: The x-values go up by 1 unit and the y-values go down by 3 units. Therefore, the slope is: −3/1 = −3. Since there is a constant rate of change, the relation in the table is linear.


Example 3
Determine whether the relation in the table below is linear or non-linear.

X  |  Y

-2  |  5

0   |  5

2   |  5

4   |  5

Solution: The x-values are going up by 2 units at a time. On the other hand, the y-values are not going up or down, that is, the change in the y-values is 0 units. Therefore, the slope is: 0/2 = 0. Since there is a constant rate of change, the relation in the table is linear.

Example 4

Determine whether the relation in the table below is linear or non-linear.

X    |  Y

0     |  3,200

1     |  3,255

10   |  3,750

100 |  8,700

Solution: Here we are comparing the number of products produced to the costs of a business. Since the x-values do not go up by the same amount, we need to consider the ratios between individual pairs of points. The difference between 1 and 0 is 1 and the difference between their respective y-values, 3,255 and 3,200, is 55. Therefore, the slope between these two points is: 55/1 = 55.

Let's consider another pair of points. The difference between 10 and 1 is 9 and the difference between their respective y-values, 3,750 and 3,255, is 495. Therefore, the slope between these two points is: 495/ 9 = 55/1 = 55.

Another pair of values gives the difference between 100 and 10 is 90 and the difference between their respective y-values, 8,700 and 3,750 is 4,950. Therefore, the slope between these two points is: 4,950/90 = 55/1 = 55.

Note, we can choose any pair of points in the table, that is, they do not have to be consecutive. Since there is a constant rate of change, the relation is linear.

Example 5

Determine whether the relation in the table below is linear or non-linear.

X  |  Y

0   |  2

1   |  3

2   |  5

3   |  8

Solution: As the x-values go up by 1, the y-values do not increase by the same amount. First, the y-values increase by 1, then 2, then 3. Since the change in the y-values is not consistent, this relation is non-linear.

Reference

CK-12: Introduction: Linear Equations and Inequalities of Two Variables

CK-12: Introduction to Linear Equations In Two Variables

Cuemath: Linear Equations

Lumen: Graphing Linear Equations

Scientific Notation

I. Intro

Scientific notation (also called Standard form in Britain) is used by scientists, mathematicians, and engineers when they are working with very large or very small numbers. Numbers are written in this format:

a x 10^b

The letter a is decimal number where 1 ≤ a < 10, and b is an integer.

When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1.

II. Converting A Number To Scientific Notation

1. Move the decimal point so that the number is rewritten and contains only ONE non-zero digit to the left of the decimal point.

2. Count how many places you moved the decimal point in Step 1. This number becomes the exponent.

3. If you moved the decimal to the left, the exponent is positive. If you moved the decimal to the right, the exponent is negative.

Example:

Write 450,000 in scientific notation

The unwritten decimal point in this number follows the last zero. Move the decimal point from this position to the left and stop just before the last digit, giving you 4.5.
The decimal point was moved five places to the left, so the exponent is a positive 5. So, in scientific notation:

450,000 = 4.5 x 10⁵.

Example:

Write 0.0000000102 in scientific notation

We moved the decimal point to the right eight places so the exponent is 8. Since we moved to the right the exponent is a negative.

1.02 x 10^-8

III. Converting From Scientific Notation To A Decimal Number

To convert from scientific notation to a decimal number, simply reverse the process. 

Example:

Write 4.5 x 10⁵ as a decimal number

The decimal is a positive number so we move the decimal five places to the right.  

4.5 x 10⁵ = 450,000

III. Conceptual Understanding

Powers of 10

1 x 10³ = 1000

1 x 10² = 100

1 x 10¹ = 10

1 x 10⁰ = 0

1 x 10^-1 = 1/10 = 0.1

1 x 10^-2 = 1/100 = 0.01

1 x 10^-3 = 1/1000 = 0.001

IXL: Scientific Notation

Mathbitsnotebook: Scientific Notation

Lumen: Scientific Notation

Math is Fun: Scientific Notation

CK-12: Scientific Notation

Khan: Scientific Notation

math typeit