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By the end of this section, you will be able to:
  • Recognize the relation between the solutions of an equation and its graph
  • Graph a linear equation by plotting points
  • Graph vertical and horizontal lines

Before you get started, take this readiness quiz.

  1. Evaluate: 3 x + 2 when x = −1 .
    If you missed this problem, review Multiply and Divide Integers .
  2. Solve the formula: 5 x + 2 y = 20 for y .
    If you missed this problem, review Solve a Formula for a Specific Variable .
  3. Simplify: 3 8 ( −24 ) .
    If you missed this problem, review Multiply and Divide Fractions .

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 3 x + 2 y = 6 . They are listed in the table below. So, the ordered pairs ( 0 , 3 ) , ( 2 , 0 ) , ( 1 , 3 2 ) , ( 4 , 3 ) , are some solutions to the equation 3 x + 2 y = 6 . 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 3 x + 2 y = 6 . 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 ( 2 , 6 ) is on the line shown in [link] . If you substitute x = 2 and y = 6 into the equation, you find that it is a solution to the equation.

...

So ( 4 , 1 ) is not a solution to the equation 3 x + 2 y = 6 . Therefore the point ( 4 , 1 ) 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 3 x + 2 y = 6 .

Graph of a linear equation

The graph of a linear equation A x + B y = C 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 y = 2 x 3 is shown below.

...

For each ordered pair decide

  1. Is the ordered pair a solution to the equation?
  2. Is the point on the line?

  1. ( 0 , 3 )
  2. ( 3 , 3 )
  3. ( 2 , 3 )
  4. ( 1 , 5 )

Substitute the x - and y -values into the equation to check if the ordered pair is a solution to the equation.


...

Plot the points A: ( 0 , 3 ) B: ( 3 , 3 ) C: ( 2 , 3 ) and D: ( 1 , 5 ) .
The points ( 0 , 3 ) , ( 3 , 3 ) , and ( 1 , 5 ) are on the line y = 2 x 3 , and the point ( 2 , 3 ) is not on the line.

...

The points which are solutions to y = 2 x 3 are on the line, but the point which is not a solution is not on the line.

Got questions? Get instant answers now!

The graph of y = 3 x 1 is shown.

For each ordered pair, decide

  1. is the ordered pair a solution to the equation?
  2. is the point on the line?
...
  1. ( 0 , 1 )
  2. ( 2 , 2 )
  3. ( 3 , 1 )
  4. ( 1 , 4 )
  1. yes yes
  2. no no
  3. no no
  4. yes yes
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 y = 2 x + 1 by plotting points.

Questions & Answers

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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