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Before you get started, take this readiness quiz.
In the previous section, we found several solutions to the equation $3x+2y=6$ . They are listed in [link] . So, the ordered pairs $\left(0,3\right)$ , $\left(2,0\right)$ , and $\left(1,\frac{3}{2}\right)$ are some solutions to the equation $3x+2y=6$ . We can plot these solutions in the rectangular coordinate system as shown in [link] .
$3x+2y=6$ | ||
$x$ | $y$ | $\left(x,y\right)$ |
0 | 3 | $\left(0,3\right)$ |
2 | 0 | $\left(2,0\right)$ |
1 | $\frac{3}{2}$ | $\left(1,\frac{3}{2}\right)$ |
Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation $3x+2y=6$ . See [link] . 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 $\left(\mathrm{-2},6\right)$ is on the line shown in [link] . If you substitute $x=\mathrm{-2}$ and $y=6$ into the equation, you find that it is a solution to the equation.
So the point $\left(\mathrm{-2},6\right)$ is a solution to the equation $3x+2y=6$ . (The phrase “the point whose coordinates are $\left(\mathrm{-2},6\right)$ ” is often shortened to “the point $\left(\mathrm{-2},6\right)$ .”)
So $\left(4,1\right)$ is not a solution to the equation $3x+2y=6$ . Therefore, the point $\left(4,1\right)$ is not on the line. See [link] . 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 a linear equation $Ax+By=C$ is a line.
The graph of $y=2x-3$ is shown.
For each ordered pair, decide:
ⓐ Is the ordered pair a solution to the equation?
ⓑ Is the point on the line?
A $\left(0,\mathrm{-3}\right)$ B $\left(3,3\right)$ C $\left(2,\mathrm{-3}\right)$ D $\left(\mathrm{-1},\mathrm{-5}\right)$
Substitute the x - and y - values into the equation to check if the ordered pair is a solution to the equation.
The points $\left(0,3\right)$ , $\left(3,3\right)$ , and $\left(\mathrm{-1},\mathrm{-5}\right)$ are on the line $y=2x-3$ , and the point $\left(2,\mathrm{-3}\right)$ is not on the line.
The points that are solutions to $y=2x-3$ are on the line, but the point that is not a solution is not on the line.
Use the graph of $y=3x-1$ to decide whether each ordered pair is:
ⓐ $\left(0,\mathrm{-1}\right)$ ⓑ $\left(2,5\right)$
ⓐ yes, yes ⓑ yes, yes
Use graph of $y=3x-1$ to decide whether each ordered pair is:
ⓐ $\left(3,\mathrm{-1}\right)$ ⓑ $\left(\mathrm{-1},\mathrm{-4}\right)$
ⓐ no, no ⓑ yes, yes
There are several methods that can be used to graph a linear equation. The method we used to graph $3x+2y=6$ is called plotting points, or the Point–Plotting Method.
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