第4章グラフ

In the previous section, we found several solutions to the equation $3x+2y=6$. They are listed in Table 4.11. 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 Figure 4.5.

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 Figure 4.6. 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 $\left(\mathrm{-2},6\right)$ is on the line shown in Figure 4.7. 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 Figure 4.6. 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 $3x+2y=6$.

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.

The steps to take when graphing a linear equation by plotting points are summarized below.

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 only plot 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. Look at the difference between part (a) and part (b) in Figure 4.8.

Let’s do another example. This time, we’ll show the last two steps all on one grid.

When an equation includes a fraction as the coefficient of $x$, we can still substitute any numbers for $x$. But the math is easier if we make ‘good’ choices for the values of $x$. This way we will avoid fraction answers, which are hard to graph precisely.

So far, all the equations we graphed had $y$ given in terms of $x$. Now we’ll graph an equation with $x$ and $y$ on the same side. Let’s see what happens in the equation $2x+y=3$. If $y=0$ what is the value of $x$?

This point has a fraction for the x- coordinate and, while we could graph this point, it is hard to be precise graphing fractions. Remember in the example $y=\frac{1}{2}x+3$, we carefully chose values for $x$ so as not to graph fractions at all. If we solve the equation $2x+y=3$ for $y$, it will be easier to find three solutions to the equation.

The solutions for $x=0$, $x=1$, and $x=\mathrm{-1}$ are shown in the Table 4.14. The graph is shown in Figure 4.9.

Can you locate the point $\left(\frac{3}{2},0\right)$, which we found by letting $y=0$, on the line?

If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x- and y-axes are the same, the graphs match!

The equation in Example 4.14 was written in standard form, with both $x$ and $y$ on the same side. We solved that equation for $y$ in just one step. But for other equations in standard form it is not that easy to solve for $y$, so we will leave them in standard form. We can still find a first point to plot by letting $x=0$ and solving for $y$. We can plot a second point by letting $y=0$ and then solving for $x$. Then we will plot a third point by using some other value for $x$ or $y$.