3B.1 Graph Linear Equations in Two Variables
3B.1 Learning Objectives
- Create a table of ordered pairs from a two-variable linear equation
- Graph a two-variable linear equation using a table of ordered pairs
- Determine whether an ordered pair is a solution of an equation
- Recognize when an ordered pair is a y-intercept or an x-intercept
- Graph a linear equation using x- and y-intercepts
Example 3b.1.A
Describe the point shown as an ordered pair.Answer: Begin at the origin and move along the x-axis. This is the x-coordinate and is written first in the ordered pair.
(5, y)
Move from 5 up to the ordered pair and read the number on the y-axis. This is the y-coordinate and is written second in the ordered pair.(5, 2)
Answer
The point shown as an ordered pair is (5, 2).Example
Plot the point [latex](−4,−2)[/latex]. The x-coordinate is [latex]−4[/latex] because it comes first in the ordered pair. Start at the origin and move 4 units in a negative direction (left) along the x-axis. The y-coordinate is [latex]−2[/latex] because it comes second in the ordered pair. Now move 2 units in a negative direction (down). If you look over to the y-axis, you should be lined up with [latex]−2[/latex] on that axis.Answer: Draw a point at this location and label the point [latex](−4,−2)[/latex].
https://youtu.be/p_MESleS3mw Graphing ordered pairs is only the beginning of the story. Once you know how to place points on a grid, you can use them to make sense of all kinds of mathematical relationships.Plotting points to graph linear relationships
x-coordinate | y-coordinate |
0 | 0 |
1 | 2 |
2 | 4 |
3 | 6 |
4 | 8 |
Example 3b.1.B
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.
x values | [latex]2x+3[/latex] | y values |
0 | 2(0) + 3 | 3 |
1 | 2(1) + 3 | 5 |
2 | 2(2) + 3 | 7 |
3 | 2(3) + 3 | 9 |
(0, 3)
(1, 5)
(2, 7)
(3, 9)
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
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.
Example 3b.1.C
Determine whether [latex](−2,4)[/latex] is a solution to the equation [latex]4y+5x=3[/latex].Answer: For this problem, you will use the substitution method. Substitute [latex]x=−2[/latex] and [latex]y=4[/latex] into the equation.
[latex]\begin{array}{r}4y+5x=3\\4\left(4\right)+5\left(−2\right)=3\end{array}[/latex]
Evaluate.[latex]\begin{array}{r}16+\left(−10\right)=3\\6=3\end{array}[/latex]
The statement is not true, so [latex](−2,4)[/latex] is not a solution to the equation [latex]4y+5x=3[/latex].Answer
[latex](−2,4)[/latex] is not a solution to the equation [latex]4y+5x=3[/latex].3B.1.1 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. The x-intercept above is the point [latex](−2,0)[/latex]. The y-intercept above is the point (0, 2). Notice that the y-intercept always occurs where [latex]x=0[/latex], and the x-intercept always occurs where [latex]y=0[/latex]. To find the x- and y-intercepts of a linear equation, you can substitute 0 for y and for x respectively. For example, the linear equation [latex]3y+2x=6[/latex] has an x intercept when [latex]y=0[/latex], so [latex]3\left(0\right)+2x=6\\[/latex].[latex]\begin{array}{r}2x=6\\x=3\end{array}[/latex]
The x-intercept is [latex](3,0)[/latex]. Likewise the y-intercept occurs when [latex]x=0[/latex].[latex]\begin{array}{r}3y+2\left(0\right)=6\\3y=6\\y=2\end{array}[/latex]
The y-intercept is [latex](0,2)[/latex].Using Intercepts to Graph Lines
You can use intercepts to graph linear equations. Once you have found the two intercepts, draw a line through them. Let’s do it with the equation [latex]3y+2x=6[/latex]. You figured out that the intercepts of the line this equation represents are [latex](0,2)[/latex] and [latex](3,0)[/latex]. That’s all you need to know.Example 3b.1.D
Graph [latex]5y+3x=30[/latex] using the x and y-intercepts.Answer: When an equation is in [latex]Ax+By=C[/latex] form, you can easily find the x- and y-intercepts and then graph.
[latex]\begin{array}{r}5y+3x=30\\5y+3\left(0\right)=30\\5y+0=30\\5y=30\\y=\,\,\,6\\y\text{-intercept}\,\left(0,6\right)\end{array}[/latex]
To find the y-intercept, set [latex]x=0[/latex] and solve for y.[latex]\begin{array}{r}5y+3x=30\\5\left(0\right)+3x=30\\0+3x=30\\3x=30\\x=10\\x\text{-intercept}\left(10,0\right)\end{array}[/latex]
To find the x-intercept, set [latex]y=0[/latex] and solve for x.Answer
Example 3b.1.E
Graph [latex]y=2x-4[/latex] using the x and y-intercepts.Answer: First, find the y-intercept. Set x equal to zero and solve for y.
[latex]\begin{array}{l}y=2x-4\\y=2\left(0\right)-4\\y=0-4\\y=-4\\y\text{-intercept}\left(0,-4\right)\end{array}[/latex]
To find the x-intercept, set [latex]y=0[/latex] and solve for x.[latex]\begin{array}{l}y=2x-4\\0=2x-4\\4=2x\\x=2\\x\text{-intercept}\left(2,0\right)\end{array}[/latex]
Answer
Licenses & Attributions
CC licensed content, Original
- Authored by: Quadrants on the Coordinate Plane. License: CC BY: Attribution.
CC licensed content, Shared previously
- Graph Basic Linear Equations by Completing a Table of Values. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Determine If an Ordered Pair is a Solution to a Linear Equation. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Plot Points Given as Ordered Pairs on the Coordinate Plane. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.