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Before you get started, take this readiness quiz.
In Use the Rectangular Coordinate System , we found a few solutions to the equation $3x+2y=6$ . They are listed in the table below. So, the ordered pairs $(0,3)$ , $(2,0)$ , $(1,\frac{3}{2})$ , $(4,-3)$ , are some solutions to the equation $3x+2y=6$ . 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 $3x+2y=6$ . 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 $3x+2y=6$ . 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 $3x+2y=6$ .
The graph of $y=2x-3$ is shown below.
For each ordered pair decide
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
$y=2x-3$ , and the point
$(2,-3)$ is not on the line.
The points which are solutions to $y=2x-3$ are on the line, but the point which is not a solution is not on the line.
The graph of $y=3x-1$ is shown.
For each ordered pair, decide
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 $y=2x+1$ by plotting points.
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