Graphing Linear Equations Using Ordered Pairs
Learning Outcomes
- Determine whether an ordered pair is a solution of an equation
- Complete a table of solutions for a linear equation
- Graph linear equations in different forms using ordered pairs
You can use a coordinate plane to plot points and to map various relationships, such as the relationship between an object’s distance and the elapsed time. Many mathematical relationships are linear relationships. Let’s look at what a linear relationship is.
A linear relationship is a relationship between variables such that when plotted on a coordinate plane, the points lie on a line. Let’s start by looking at a series of points in Quadrant I on the coordinate plane.
Look at the five ordered pairs (and their x- and y-coordinates) below. Do you see any pattern to the location of the points? If this pattern continued, what other points could be on the line?
You probably identified that if this pattern continued the next ordered pair would be at . This makes sense because the point “lines up” with the other points in the series—it is literally on the same line as the others. Applying the same logic, you may identify that the ordered pairs and would also belong if this coordinate plane were larger; they, too, will line up with the other points.
These series of points can also be represented in a table. In the table below, the x- and y-coordinates of each ordered pair on the graph is recorded.
Plotting points to graph linear relationships
x-coordinate | y-coordinate |
Example
Graph the linear equation .Answer: Evaluate for different values of x, and create a table of corresponding x and y values.
Draw a line through the points to indicate all of the points on the line.
Answer
Graph the linear equation
https://youtu.be/f5yvGPEWpvEExample
Graph the linear equation .Answer: Evaluate for different values of x, and create a table of corresponding x and y values.
Convert the table to ordered pairs. Plot the ordered pairs. Draw a line through the points to indicate all of the points on the line.
Answer
Try It
[ohm_question]92754[/ohm_question]Ordered Pairs as Solutions
So far, you have considered the following ideas about lines: a line is a visual representation of a linear equation, and the line itself is made up of an infinite number of points (or ordered pairs). The picture below shows the line of the linear equation with some of the specific points on the line. Every point on the line is a solution to the equation . You can try any of the points that are labeled like the ordered pair, .You can also try ANY of the other points on the line. Every point on the line is a solution to the equation . All this means is that determining whether an ordered pair is a solution of an equation is pretty straightforward. If the ordered pair is on the line created by the linear equation, then it is a solution to the equation. But if the ordered pair is not on the line—no matter how close it may look—then it is not a solution to the equation.
Identifying Solutions
To find out whether an ordered pair is a solution of a linear equation, you can do the following:- Graph the linear equation, and graph the ordered pair. If the ordered pair appears to be on the graph of a line, then it is a possible solution of the linear equation. If the ordered pair does not lie on the graph of a line, then it is not a solution.
- Substitute the (x, y) values into the equation. If the equation yields a true statement, then the ordered pair is a solution of the linear equation. If the ordered pair does not yield a true statement then it is not a solution.
Try It
[ohm_question]15613[/ohm_question]Watch the video below to see more about how solutions to linear equations lie on their graphs.
https://youtu.be/pJtxugdFjEkSolve for y, then graph a linear equation
The linear equations we have graphed so far are in the form where m and b are real numbers. In this section we will graph linear equations that appear in different forms than we have seen. TIP: You can use ANY values for x that fit on your graph! Pick easy numbers to work with, too!Example
Graph the linear equation .Answer: First, solve for , then the equation will look familiar and you can create a table of ordered pairs.
Evaluate for different values of , and create a table of corresponding and values. TIP: You can use ANY values for x that fit on the graph!
Draw a line through the points to indicate all of the points on the line.
Answer
Video: Solve for y, then graph a linear equation
https://youtu.be/6yL3gfPbOt8Horizontal and Vertical Lines
The linear equations and only have one variable in each of them. However, because these are linear equations, then they will graph on a coordinate plane just as the linear equations above do. Just think of the equation as and think of as .Example
Graph .Answer:
Plot the ordered pairs (shown below). Draw a line through the points to indicate all of the points on the line.
Answer
Notice that graphs as a horizontal line.Watch the video below for more examples of how to graph horizontal and vertical lines.
https://youtu.be/2A2fhImjOBcContribute!
Did you have an idea for improving this content? We’d love your input.