# Graphs

## Graph  Linear Equations in Two Variables


### Recognize the Relationship Between the Solutions of an Equation and its Graph

In the previous section, we found several solutions to the equation . They are listed in . So, the ordered pairs , , and  are some solutions to the equation . We can plot these solutions in the rectangular coordinate system as shown in .
Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation . See . Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.
Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions.
Notice that the point whose coordinates are  is on the line shown in . If you substitute  and  into the equation, you find that it is a solution to the equation.
So the point  is a solution to the equation . (The phrase “the point whose coordinates are ” is often shortened to “the point .”)
So  is not a solution to the equation . Therefore, the point  is not on the line. See . This is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to the 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 .

### Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The method we used to graph  is called plotting points, or the Point–Plotting Method.
The steps to take when graphing a linear equation by plotting points are summarized below.
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 part (a) and part (b) in .
Let’s do another example. This time, we’ll show the last two steps all on one grid.
When an equation includes a fraction as the coefficient of , we can still substitute any numbers for . But the math is easier if we make ‘good’ choices for the values of . This way we will avoid fraction answers, which are hard to graph precisely.
So far, all the equations we graphed had  given in terms of . Now we’ll graph an equation with  and  on the same side. Let’s see what happens in the equation . If  what is the value of ?
This point has a fraction for the x- coordinate and, while we could graph this point, it is hard to be precise graphing fractions. Remember in the example , we carefully chose values for  so as not to graph fractions at all. If we solve the equation  for , it will be easier to find three solutions to the equation.
The solutions for , , and  are shown in the . The graph is shown in .
Can you locate the point , which we found by letting , on the line?
If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x- and y-axis are the same, the graphs match!
The equation in  was written in standard form, with both  and  on the same side. We solved that equation for  in just one step. But for other equations in standard form it is not that easy to solve for , so we will leave them in standard form. We can still find a first point to plot by letting  and solving for . We can plot a second point by letting  and then solving for . Then we will plot a third point by using some other value for  or .

### Graph Vertical and Horizontal Lines

Can we graph an equation with only one variable? Just  and no , or just  without an ? How will we make a table of values to get the points to plot?
Let’s consider the equation . This equation has only one variable, . The equation says that  is always equal to , so its value does not depend on . No matter what  is, the value of  is always .
So to make a table of values, write  in for all the  values. Then choose any values for . Since  does not depend on , 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 .
Plot the points from  and connect them with a straight line. Notice in  that we have graphed a vertical line.
 
What if the equation has  but no ? Let’s graph the equation . This time the y- value is a constant, so in this equation,  does not depend on . Fill in 4 for all the ’s in  and then choose any values for . We’ll use 0, 2, and 4 for the x-coordinates.
The graph is a horizontal line passing through the y-axis at 4. See .
 
The equations for vertical and horizontal lines look very similar to equations like  What is the difference between the equations  and ?
The equation  has both  and . The value of  depends on the value of . The y-coordinate changes according to the value of . The equation  has only one variable. The value of  is constant. The y-coordinate is always 4. It does not depend on the value of . See .
Notice, in , the equation  gives a slanted line, while  gives a horizontal line.

### Key Concepts

1. Graph a Linear Equation by Plotting Points



### Practice Makes Perfect

Recognize the Relationship Between the Solutions of an Equation and its Graph
In the following exercises, for each ordered pair, decide:
ⓐ Is the ordered pair a solution to the equation? ⓑ Is the point on the line?
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.

### Mixed Practice

In the following exercises, graph each equation.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals?