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
Plotting points to graph linear relationships
x-coordinate | y-coordinate |
[latex]0[/latex] | [latex]0[/latex] |
[latex]1[/latex] | [latex]2[/latex] |
[latex]2[/latex] | [latex]4[/latex] |
[latex]3[/latex] | [latex]6[/latex] |
[latex]4[/latex] | [latex]8[/latex] |
Example
Graph the linear equation [latex]y=−1.5x[/latex].Answer: Evaluate [latex]y=−1.5x[/latex] for different values of x, and create a table of corresponding x and y values.
[latex]x values[/latex] | [latex]−1.5x[/latex] | [latex]y values[/latex] |
[latex]0[/latex] | [latex]−1.5(0)[/latex] | [latex]0[/latex] |
[latex]2[/latex] | [latex]−1.5(2)[/latex] | [latex]−3[/latex] |
[latex]4[/latex] | [latex]−1.5(4)[/latex] | [latex]−6[/latex] |
[latex]6[/latex] | [latex]−1.5(6)[/latex] | [latex]−9[/latex] |
[latex](0,0)[/latex]
[latex](2,−3)[/latex]
[latex](4,−6)[/latex]
[latex](6,−9)[/latex]
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 [latex]y=2x+3[/latex].Answer: Evaluate [latex]y=2x+3[/latex] for different values of x, and create a table of corresponding x and y values.
[latex]x values[/latex] | [latex]2x+3[/latex] | [latex]y values[/latex] |
[latex]0[/latex] | [latex]2(0) + 3[/latex] | [latex]3[/latex] |
[latex]1[/latex] | [latex]2(1) + 3[/latex] | [latex]5[/latex] |
[latex]2[/latex] | [latex]2(2) + 3[/latex] | [latex]7[/latex] |
[latex]3[/latex] | [latex]2(3) + 3[/latex] | [latex]9[/latex] |
[latex](0, 3)[/latex]
[latex](1, 5)[/latex]
[latex](2, 7)[/latex]
[latex](3, 9)[/latex]
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 [latex]y=2x–5[/latex] with some of the specific points on the line. Every point on the line is a solution to the equation [latex]y=2x–5[/latex]. You can try any of the points that are labeled like the ordered pair, [latex](1,−3)[/latex].[latex]\begin{array}{l}\,\,\,\,y=2x-5\\-3=2\left(1\right)-5\\-3=2-5\\-3=-3\\\text{This is true.}\end{array}[/latex]
You can also try ANY of the other points on the line. Every point on the line is a solution to the equation [latex]y=2x–5[/latex]. 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 [latex]y=mx+b[/latex] 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 [latex]y+3x=5[/latex].Answer: First, solve [latex]y+3x=5[/latex] for [latex]y[/latex], then the equation will look familiar and you can create a table of ordered pairs.
[latex]\begin{array}{r}y+3x-3x=5-3x\\y=5-3x\end{array}[/latex]
Evaluate [latex]y=5–3x[/latex] for different values of [latex]x[/latex], and create a table of corresponding [latex]x[/latex] and [latex]y[/latex] values. TIP: You can use ANY values for x that fit on the graph![latex]x-values[/latex] | [latex]5–3x[/latex] | [latex]y-values[/latex] |
[latex]0[/latex] | [latex]5–3(0)[/latex] | [latex]5[/latex] |
[latex]1[/latex] | [latex]5–3(1)[/latex] | [latex]2[/latex] |
[latex]2[/latex] | [latex]5–3(2)[/latex] | [latex]−1[/latex] |
[latex]3[/latex] | [latex]5–3(3)[/latex] | [latex]−4[/latex] |
[latex](0,5)[/latex]
[latex](1,2)[/latex]
[latex](2,−1)[/latex]
[latex](3,−4)[/latex]
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 [latex]x=2[/latex] and [latex]y=−3[/latex] 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 [latex]x=2[/latex] as [latex]x=0y+2[/latex] and think of [latex]y=−3[/latex] as [latex]y=0x–3[/latex].Example
Graph [latex]y=−3[/latex].Answer:
[latex]x-values[/latex] | [latex]0x–3[/latex] | [latex]y-values[/latex] |
[latex]0[/latex] | [latex]0(0)–3[/latex] | [latex]−3[/latex] |
[latex]1[/latex] | [latex]0(1)–3[/latex] | [latex]−3[/latex] |
[latex]2[/latex] | [latex]0(2)–3[/latex] | [latex]−3[/latex] |
[latex]3[/latex] | [latex]0(3)–3[/latex] | [latex]−3[/latex] |
[latex](0,−3)[/latex]
[latex](1,−3)[/latex]
[latex](2,−3)[/latex]
[latex](3,−3)[/latex]
Plot the ordered pairs (shown below). Draw a line through the points to indicate all of the points on the line.Answer
Notice that [latex]y=−3[/latex] graphs as a horizontal line.Watch the video below for more examples of how to graph horizontal and vertical lines.
https://youtu.be/2A2fhImjOBcContribute!
Licenses & Attributions
CC licensed content, Shared previously
- Graph Basic Linear Equations by Completing a Table of Values. Authored by: mathispower4u. License: All Rights Reserved. License terms: Standard YouTube License.
- Determine If an Ordered Pair is a Solution to a Linear Equation. Authored by: mathispower4u. License: All Rights Reserved. License terms: Standard YouTube License.