Linear Equations in Two Variables Lesson

Up to this point, we have exclusively worked with equations that contain only one variable. These equations are known as linear equations in one variable. In most cases, a linear equation in one variable will have exactly one solution.
3x + 14 =  35
x = 7
For our above equation, the only value that can replace the variable x and result in a true statement is 7. Graphically, this type of solution can be displayed as a single point on a horizontal number line.
x = 7

Linear Equation in Two Variables Definition

Let's now think about a different type of equation, one with two variables involved. A linear equation in two variables is of the form:
ax + by = c
This is the format we see in our textbook, with x and y as our variables. We have a and b as the coefficients of x and y, whereas c is a constant. We can replace a, b, and c with any real numbers as long as a and b are not both zero. Let's look at a few examples of a linear equation in two variables.

• 3x + 2y = 19
• 6x - y = 13
• x + 11y = 94
• -12x + 20y = 49

When we work with a linear equation in two variables, we generally have an infinite number of solutions. For this reason, it is normal to graph the equation and give a visual representation of the solution set. Before we can jump in and start graphing linear equations in two variables, we need to learn a few things first.

What is an Ordered Pair (x,y)?

When we write the solution for a linear equation in two variables, it is generally in the format of an ordered pair. This ordered pair is given as:
(x,y)
This means the first value (leftmost) is to be plugged in for x and the second value (rightmost) is to be plugged in for y.
How do we obtain an ordered pair? We can pick a value for x and solve for y or pick a value for y and solve for x. Let's suppose we have the following equation:
3x - y = 10
Let's suppose we choose a value for x that is 0. In other words, we would have the ordered pair:
(0, __)
To fill in the blank or find the corresponding y-value, we plug in a 0 for x and solve for y:
3(0) - y = 10
0 - y = 10
-y = 10
y = -10
(0, -10)
One such solution for this equation is the ordered pair (0,-10). This means that when x is 0 and y is -10, we will get a true statement.
3(0) - (-10) = 10
0 + 10 = 10
10 = 10
Let's look at an example.
Example 1: Complete each table of values.
-2x - 7y = 28

x	y
__	0
0	__
__	-6

For our first ordered pair, we are given a y-value of 0. To find the unknown x-value, we plug in a 0 for y and solve for x:
-2x - 7(0) = 28
-2x = 28
x = -14
(-14, 0)
Let's update our table:

x	y	(x, y)
-14	0	(-14, 0)
0	__
__	-6

For our second ordered pair, we are given an x-value of 0. To find the unknown y-value, we plug in a 0 for x and solve for y:
-2(0) - 7y = 28
0 - 7y = 28
-7y = 28
y = -4
(0, -4)
Let's update our table:

x	y	(x, y)
-14	0	(-14, 0)
0	-4	(0, -4)
__	-6

For our third ordered pair, we are given a y-value of -6. To find the unknown x-value, we plug in a -6 for y and solve for x:
-2x - 7(-6) = 28
-2x + 42 = 28
-2x = -14
x = 7
(7, -6)
Let's update our table:

x	y	(x, y)
-14	0	(-14, 0)
0	-4	(0, -4)
7	-6	(7, -6)

Cartesian Coordinate System (Rectangular Coordinate Plane)

The Cartesian Coordinate System is a formal name for a rectangular coordinate plane. This plane allows us to plot ordered pairs and graph linear equations in two variables. The name for the Cartesian Coordinate System may vary. We may hear this called the rectangular coordinate plane, the coordinate plane, the coordinate system, the Cartesian coordinate plane,...etc. For the purposes of this course, we will normally refer to this as the coordinate plane. To draw the coordinate plane, we set up two number lines. The x-axis is the horizontal number line, it represents x-values. The y-axis is the vertical number line, it represents y-values. Let's look at the coordinate plane: The point at which the x-axis and y-axis intersect is known as the origin. At the origin, both the x-value and y-value are 0, as an ordered pair: (0,0). On our horizontal number line or x-axis, numbers increase as we move to the right and decrease as we move to the left. On the vertical number line or y-axis, numbers increase as we move up and decrease as we move down. The coordinate plane is split up into four quadrants labeled with Roman numerals as I, II, III, and IV. Quadrant I is located in the upper right quadrant. The additional three quadrants are found moving counterclockwise. It is important to note that a point on the x-axis or y-axis does not belong in any quadrant.
Quadrant I » (+, +)
Quadrant II » (-, +)
Quadrant III » (-, -)
Quadrant IV » (+, -)

How to Plot an Ordered Pair

We will often refer to an ordered pair as a "point". To plot a point (an ordered pair), we simply find the meeting point of the x-value and y-value. There are many ways to perform this operation. One method is to start at the origin, or the point (0,0). We then move by x-units left or right and y-units up or down. If the x-value is positive, this represents a movement to the right of 0 on the x-axis. If the x-value is negative, this represents a movement to the left of 0 on the x-axis. If the y-value is positive, this represents a movement above 0 on the y-axis. If the y-value is negative, this represents a movement below 0 on the y-axis. We may be given instructions to plot the point (3, 6). This means we want to locate the meeting point of an x-value of 3 and a y-value of 6. At this spot, we will draw a filled-in circle. There are many ways to find the point (3, 6). Starting from the origin (0,0) we can move 3 units right and 6 units up. This puts us at the point (3,6). Let's look at a few examples.
Example 2: Plot each ordered pair, determine its quadrant.
(-4,-3)
Starting from the origin, we can move 4 units to the left and 3 units down. Since both x and y values are negative, this point is in quadrant III. Example 3: Plot each ordered pair.
(-9,6)
Starting from the origin, we can move 9 units to the left and 6 units up. Since the x-value is negative and the y-value is positive, this point is in quadrant II.

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