4.2 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 $3x+2y=6$ . 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] .

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 $y=2x-3$ 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 $\left(\mathrm{-1},\mathrm{-5}\right)$

Solution

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

1. ⓐ
   

   

2. ⓑ Plot the points A $\left(0,3\right)$ , B $\left(3,3\right)$ , C $\left(2,\mathrm{-3}\right)$ , and D $\left(\mathrm{-1},\mathrm{-5}\right)$ .
   

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.

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Use the graph of $y=3x-1$ to decide whether each ordered pair is:

• a solution to the equation.
• on the line.

ⓐ $\left(0,\mathrm{-1}\right)$   ⓑ $\left(2,5\right)$

ⓐ yes, yes  ⓑ yes, yes

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Use graph of $y=3x-1$ to decide whether each ordered pair is:

• a solution to the equation
• on the line

ⓐ $\left(3,\mathrm{-1}\right)$   ⓑ $\left(\mathrm{-1},\mathrm{-4}\right)$

ⓐ no, no  ⓑ yes, yes

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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 $3x+2y=6$ is called plotting points, or the Point–Plotting Method.