# Graphs

## Graphing Linear Equations


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

In Use the Rectangular Coordinate System, we found a few solutions to the equation . They are listed in the table below.  So, the ordered pairs , , , , are some solutions to the equation. We can plot these solutions in the rectangular coordinate system as shown on the graph at right.





Notice how the points line up perfectly?  We connect the points with a straight line to get the graph of the equation .  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   is not a solution to the equation  . Therefore the point  is not on the line.



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 at the start of this section to graph is called plotting points, or the Point-Plotting Method.
Let’s graph the equation  by plotting points. 
We start by finding three points that are solutions to the equation. We can choose any value for  or  and then solve for the other variable.
Since  is isolated on the left side of the equation, it is easier to choose values for  We will use  and  for  for this example. We substitute each value of  into the equation and solve for 
We can organize the solutions in a table. See .
Now we plot the points on a rectangular coordinate system. Check that the points line up. If they did not line up, it would mean we made a mistake and should double-check all our work. See .
Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. The line is the graph of 
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 plot only 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. See .
When an equation includes a fraction as the coefficient of  we can 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.
In the previous example, the three points we found were easy to graph. But this is not always the case. Let’s see what happens in the equation  If  is  what is the value of 
The solution is the point  This point has a fraction for the -coordinate. 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.
Now we can choose values for  that will give coordinates that are integers. The solutions for  and  are shown.

### 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  The equation says that  is always equal to  so its value does not depend on  No matter what  is, the value of  is always 
To make a table of solutions, we write  for all the  values. Then choose any values for  Since  does not depend on  you can chose any numbers you like. But to fit the size of our coordinate graph, we’ll use  and  for the -coordinates as shown in the table.
Then plot the points and connect them with a straight line. Notice in  that the graph is a vertical line.
What if the equation has  but no ? Let’s graph the equation  This time the -value is a constant, so in this equation  does not depend on 
To make a table of solutions, write  for all the  values and then choose any values for 
We’ll use  and  for the -values.
Plot the points and connect them, as shown in . This graph is a horizontal line passing through the  at 
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  changes according to the value of 
The equation  has only one variable. The value of  is constant. The  is always  It does not depend on the value of 
The graph shows both equations.
Notice that the equation  gives a slanted line whereas  gives a horizontal line.

### Key Concepts

1. Graph a linear equation by plotting points.
2. Graph of a Linear Equation:The graph of a linear equation  is a straight line.

### Practice Makes Perfect

Recognize the Relation Between the Solutions of an Equation and its Graph 
In each of the following exercises, an equation and its graph is shown. For each ordered pair, decide

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 the vertical and horizontal lines.
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 objectives?
