Graphs and Functions
Graph Linear Equations in Two Variables
Learning Objectives
By the end of this section, you will be able to:
- Plot points in a rectangular coordinate system
- Graph a linear equation by plotting points
- Graph vertical and horizontal lines
- Find the x- and y-intercepts
- Graph a line using the intercepts
Before you get started, take this readiness quiz.
Plot Points on a Rectangular Coordinate System
Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the “coordinate plane.”
The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See (Figure).
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point. The phrase “ordered pair” means that the order is important.
An ordered pair, gives the coordinates of a point in a rectangular coordinate system. The first number is the x-coordinate. The second number is the y-coordinate.
What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is The point has a special name. It is called the origin.
The point is called the origin. It is the point where the x-axis and y-axis intersect.
We use the coordinates to locate a point on the xy-plane. Let’s plot the point as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through Then, locate 3 on the y-axis and sketch a horizontal line through Now, find the point where these two lines meet—that is the point with coordinates See (Figure).
Notice that the vertical line through and the horizontal line through are not part of the graph. We just used them to help us locate the point
When one of the coordinate is zero, the point lies on one of the axes. In (Figure) the point is on the y-axis and the point is on the x-axis.
Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates
Points with an x-coordinate equal to 0 are on the y-axis, and have coordinates
Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:
ⓐⓑⓒⓓⓔ
The first number of the coordinate pair is the x-coordinate, and the second number is the y-coordinate. To plot each point, sketch a vertical line through the x-coordinate and a horizontal line through the y-coordinate. Their intersection is the point.
ⓐ Since the point is to the left of the y-axis. Also, since the point is above the x-axis. The point is in Quadrant II.
ⓑ Since the point is to the left of the y-axis. Also, since the point is below the x-axis. The point is in Quadrant III.
ⓒ Since the point is to the right of the y-axis. Since the point is below the x-axis. The point is in Quadrant IV.
ⓓ Since the point whose coordinates are is on the y-axis.
ⓔ Since the point is to the right of the y-axis. Since the point is above the x-axis. (It may be helpful to write as a mixed number or decimal.) The point is in Quadrant I.
Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:
ⓐ ⓑ ⓒ ⓓ ⓔ
Plot each point in a rectangular coordinate system and identify the quadrant in which the point is located:
ⓐ ⓑ ⓒ ⓓ ⓔ
The signs of the x-coordinate and y-coordinate affect the location of the points. You may have noticed some patterns as you graphed the points in the previous example. We can summarize sign patterns of the quadrants in this way:
Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. But equations can have more than one variable. Equations with two variables may be of the form An equation of this form is called a linear equation in two variables.
An equation of the form where A and B are not both zero, is called a linear equation in two variables.
Here is an example of a linear equation in two variables, x and y.
The equation is also a linear equation. But it does not appear to be in the form We can use the Addition Property of Equality and rewrite it in form.
By rewriting as we can easily see that it is a linear equation in two variables because it is of the form When an equation is in the form we say it is in standard form of a linear equation.
A linear equation is in standard form when it is written
Most people prefer to have A, B, and C be integers and when writing a linear equation in standard form, although it is not strictly necessary.
Linear equations have infinitely many solutions. For every number that is substituted for x there is a corresponding y value. This pair of values is a solution to the linear equation and is represented by the ordered pair When we substitute these values of x and y into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.
An ordered pair is a solution of the linear equation if the equation is a true statement when the x– and y-values of the ordered pair are substituted into the equation.
Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.
A graph is a visual representation of all the solutions of the equation. It is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to that equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation. Points not on the line are not solutions!
The graph of a linear equation is a straight line.
- Every point on the line is a solution of the equation.
- Every solution of this equation is a point on this line.
The graph of is shown.
For each ordered pair, decide:
ⓐ Is the ordered pair a solution to the equation?
ⓑ Is the point on the line?
A: B: C: D:
Substitute the x– and y-values into the equation to check if the ordered pair is a solution to the equation.
ⓐ
ⓑ Plot the points and
The points and are on the line and the point is not on the line.
The points that are solutions to are on the line, but the point that is not a solution is not on the line.
Use graph of For each ordered pair, decide:
ⓐ Is the ordered pair a solution to the equation?
ⓑ Is the point on the line?
A B
ⓐ yes, yes ⓑ yes, yes
Use graph of For each ordered pair, decide:
ⓐ Is the ordered pair a solution to the equation?
ⓑ Is the point on the line?
A B
ⓐ no, no ⓑ yes, yes
Graph a Linear Equation by Plotting Points
There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.
Graph the equation by plotting points.
Graph the equation by plotting points:
Graph the equation by plotting points:
The steps to take when graphing a linear equation by plotting points are summarized here.
- Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.
It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.
If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between these illustrations.
When an equation includes a fraction as the coefficient of , we can still substitute any numbers for x. But the arithmetic is easier if we make “good” choices for the values of x. This way we will avoid fractional answers, which are hard to graph precisely.
Graph the equation:
Find three points that are solutions to the equation. Since this equation has the fraction as a coefficient of x, we will choose values of x carefully. We will use zero as one choice and multiples of 2 for the other choices. Why are multiples of two a good choice for values of x? By choosing multiples of 2 the multiplication by simplifies to a whole number
The points are shown in (Figure).
| x | y | |
| 0 | 3 | |
| 2 | 4 | |
| 4 | 5 |
Plot the points, check that they line up, and draw the line.
Graph the equation:
Graph the equation:
Graph Vertical and Horizontal Lines
Some linear equations have only one variable. They may have just x and no y, or just y without an x. This changes how we make a table of values to get the points to plot.
Let’s consider the equation This equation has only one variable, x. The equation says that x is always equal to so its value does not depend on y. No matter what is the value of y, the value of x is always
So to make a table of values, write in for all the x-values. Then choose any values for y. Since x does not depend on y, you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See (Figure).
| x | y | |
| 1 | ||
| 2 | ||
| 3 |
Plot the points from the table and connect them with a straight line. Notice that we have graphed a vertical line.
What if the equation has y but no x? Let’s graph the equation This time the y-value is a constant, so in this equation, y does not depend on x. Fill in 4 for all the y’s in (Figure) and then choose any values for x. We’ll use 0, 2, and 4 for the x-coordinates.
| x | y | |
| 0 | 4 | |
| 2 | 4 | |
| 4 | 4 |
In this figure, we have graphed a horizontal line passing through the y-axis at 4.
A vertical line is the graph of an equation of the form
The line passes through the x-axis at
A horizontal line is the graph of an equation of the form
The line passes through the y-axis at
Graph: ⓐ ⓑ
ⓐ The equation has only one variable, x, and x is always equal to 2. We create a table where x is always 2 and then put in any values for y. The graph is a vertical line passing through the x-axis at 2.
| x | y | |
| 2 | 1 | |
| 2 | 2 | |
| 2 | 3 |
ⓑ Similarly, the equation has only one variable, y. The value of y is constant. All the ordered pairs in the next table have the same y-coordinate. The graph is a horizontal line passing through the y-axis at
| x | y | |
| 0 | ||
| 3 | ||
Graph the equations: ⓐ ⓑ
ⓐ
ⓑ
Graph the equations: ⓐ ⓑ
ⓐ
ⓑ
What is the difference between the equations and
The equation has both x and y. The value of y depends on the value of x, so the y -coordinate changes according to the value of x. The equation has only one variable. The value of y is constant, it does not depend on the value of x, so the y-coordinate is always 4.
Notice, in the graph, the equation gives a slanted line, while gives a horizontal line.
Graph and in the same rectangular coordinate system.
We notice that the first equation has the variable x, while the second does not. We make a table of points for each equation and then graph the lines. The two graphs are shown.
Graph the equations in the same rectangular coordinate system: and
Graph the equations in the same rectangular coordinate system: and
Find x– and y-intercepts
Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.
At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x-axis and the y-axis. These points are called the intercepts of a line.
The points where a line crosses the x-axis and the y-axis are called the intercepts of the line.
Let’s look at the graphs of the lines.
First, notice where each of these lines crosses the x-axis. See (Figure).
Now, let’s look at the points where these lines cross the y-axis.
| Figure | The line crosses |
|---|
the x-axis at:Ordered pair
for this pointThe line crosses
the y-axis at:Ordered pair
for this pointFigure (a)36Figure (b)4Figure (c)5Figure (d)00General Figureab
Do you see a pattern?
For each line, the y-coordinate of the point where the line crosses the x-axis is zero. The point where the line crosses the x-axis has the form and is called the x-intercept of the line. The x-intercept occurs when y is zero.
In each line, the x–coordinate of the point where the line crosses the y-axis is zero. The point where the line crosses the y-axis has the form and is called the y-intercept of the line. The y-intercept occurs when x is zero.
The x-intercept is the point where the line crosses the x-axis.
The y-intercept is the point where the line crosses the y-axis.
Find the x– and y-intercepts on each graph shown.
ⓐ The graph crosses the x-axis at the point The x-intercept is
The graph crosses the y-axis at the point The y-intercept is
ⓑ The graph crosses the x-axis at the point The x-intercept is
The graph crosses the y-axis at the point The y-intercept is
ⓒ The graph crosses the x-axis at the point The x-intercept is
The graph crosses the y-axis at the point The y-intercept is
Find the x– and y-intercepts on the graph.
x-intercept:
y-intercept:
Find the x– and y-intercepts on the graph.
x-intercept:
y-intercept:
Recognizing that the x-intercept occurs when y is zero and that the y-intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x-intercept, let and solve for x. To find the y-intercept, let and solve for y.
Use the equation of the line. To find:
- the x-intercept of the line, let and solve for x.
- the y-intercept of the line, let and solve for y.
Find the intercepts of
We will let to find the x-intercept, and let to find the y-intercept. We will fill in a table, which reminds us of what we need to find.
| To find the x-intercept, let | |
| Let | |
| Simplify. | |
| The x-intercept is: | |
| To find the y-intercept, let | |
| Let | |
| Simplify. | |
| The y-intercept is: |
The intercepts are the points and as shown in the table.
| x | y |
| 4 | 0 |
| 0 | 8 |
Find the intercepts:
x-intercept:
y-intercept:
Find the intercepts:
x-intercept:
y-intercept:
Graph a Line Using the Intercepts
To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.
Graph using the intercepts.
Graph using the intercepts:
Graph using the intercepts:
The steps to graph a linear equation using the intercepts are summarized here.
- Find the x– and y-intercepts of the line.
- Let and solve for x.
- Let and solve for y.
- Find a third solution to the equation.
- Plot the three points and check that they line up.
- Draw the line.
Graph using the intercepts.
Find the intercepts and a third point.
We list the points in the table and show the graph.
| x | y | |
| 3 | 0 | |
| 0 | ||
| 6 | 4 |
Graph using the intercepts:
Graph using the intercepts:
When the line passes through the origin, the x-intercept and the y-intercept are the same point.
Graph using the intercepts.
This line has only one intercept. It is the point
To ensure accuracy, we need to plot three points. Since the x– and y-intercepts are the same point, we need two more points to graph the line.
The resulting three points are summarized in the table.
| x | y | |
| 0 | 0 | |
| 1 | 5 | |
Plot the three points, check that they line up, and draw the line.
Graph using the intercepts:
Graph the intercepts:
Key Concepts
- Points on the Axes
- Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates
- Points with an x-coordinate equal to are on the y-axis, and have coordinates
- Quadrant
- Graph of a Linear Equation: The graph of a linear equation is a straight line.
Every point on the line is a solution of the equation.
Every solution of this equation is a point on this line.
- How to graph a linear equation by plotting points.
- Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.
- x-intercept and y-intercept of a Line
- The x-intercept is the point where the line crosses the x-axis.
- The y-intercept is the point where the line crosses the y-axis.
- Find the x– and y-intercepts from the Equation of a Line
- Use the equation of the line. To find:
the x-intercept of the line, let and solve for x.
the y-intercept of the line, let and solve for y.
- Use the equation of the line. To find:
- How to graph a linear equation using the intercepts.
- Find the x– and y-intercepts of the line.
Let and solve for x.
Let and solve for y.
- Find a third solution to the equation.
- Plot the three points and check that they line up.
- Draw the line
- Find the x– and y-intercepts of the line.
Practice Makes Perfect
Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
ⓐⓑⓒⓓ
ⓔ
ⓐⓑⓒⓓ
ⓔ
ⓐⓑⓒⓓ
ⓔ
ⓐⓑⓒⓓ
ⓔ
In the following exercises, for each ordered pair, decide
ⓐ is the ordered pair a solution to the equation? ⓑ is the point on the line?
A: B: C: D:
ⓐ A: yes, B: no, C: yes, D: yes ⓑ A: yes, B: no, C: yes, D: yes
A: B: C: D:
A: B: C: D:
ⓐ A: yes, B: yes, C: yes, D: no ⓑ A: yes, B: yes, C: yes, D: no
A: B: C: D:
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines
In the following exercises, graph each equation.
ⓐⓑ
ⓐ
ⓑ
ⓐⓑ
ⓐⓑ
ⓐ
ⓑ
ⓐⓑ
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
and
and
and
and
Find x- and y-Intercepts
In the following exercises, find the x– and y-intercepts on each graph.
In the following exercises, find the intercepts for each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
Mixed Practice
In the following exercises, graph each equation.
Writing Exercises
Explain how you would choose three x-values to make a table to graph the line
Answers will vary.
What is the difference between the equations of a vertical and a horizontal line?
Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation Why?
Answers will vary.
Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation Why?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
Confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
With some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
No, I don’t get it. This is a warning sign and you must address it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.
Glossary
- horizontal line
- A horizontal line is the graph of an equation of the form The line passes through the y-axis at
- intercepts of a line
- The points where a line crosses the x-axis and the y-axis are called the intercepts of the line.
- linear equation
- An equation of the form where A and B are not both zero, is called a linear equation in two variables.
- ordered pair
- An ordered pair, gives the coordinates of a point in a rectangular coordinate system. The first number is the x-coordinate. The second number is the y-coordinate.
- origin
- The point is called the origin. It is the point where the x-axis and y-axis intersect.
- solution of a linear equation in two variables
- An ordered pair is a solution of the linear equation if the equation is a true statement when the x– and y-values of the ordered pair are substituted into the equation.
- standard form of a linear equation
- A linear equation is in standard form when it is written
- vertical line
- A vertical line is the graph of an equation of the form The line passes through the x-axis at