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 are real numbers, 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