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The Line Defined

A linear equation is a special kind of function. Remember in the preceding section, when we defined a function by a set of ordered pairs or points? Each of those points is an individual solution for the equation y = 2x. The complete solution is all the points that satisfy the equation. If you began plotting individual solutions to a linear equation, an interesting pattern would emerge.

Look at the table for the linear equation y = 2x again. The table offers five individual solutions to the equation. Before you look at Figure 1.9, plot these five points and see if you can figure out the pattern. If you're not sure, try some additional solutions. As you plot more and more solutions, a line starts to emerge, as shown in Figure 1.9. This line is the complete solution for the linear equation y = 2x.

Figure 1.9. The complete solution for the linear equation y = 2x.

graphics/01fig09.gif

So what is the pattern? That's right—a line. That's precisely why it's called a linear equation.

The graph of an equation of the form Ax + By = C, where A and B are not both 0, is a straight line.

Conversely, every straight line is an equation of the form Ax + By = C, where A and B are not both 0.


In general, the easiest way to graph a linear equation is to transform the equation so that the y is alone on one side. Then choose a value for x, substitute it in the equation, and find a value for y. Although two ordered pairs are enough to determine the graph of a line, it's better to use a third point as a check.

TIP

Be smart when choosing values of x; keep them small and manageable. It's usually helpful to use 0 as one of your x values.


Example 1.3: Graphing a Line

Graph the equation 3x – 2y = 8.

Solution

  1. Transform the equation to get y alone on one side:

    3x – 2y = 8

    –2y = –3x + 8

    y = (3/2)x – 4

  2. It's much easier to graph points with integer coordinates, so choose x values such as 0, 2, and 4. When you plug these three values into the equation, you get the following three ordered pairs as individual solutions:

    (0,–4), (2,–1), (4,2)

  3. Graph these three points, and draw the line connecting them. (Technically, you need only two points to determine the line, but the third point serves as a double check.)

    The graph of the line 3x – 2y = 8 is shown in Figure 1.10.

    Figure 1.10. The complete solution for the linear equation 3x – 2y = 8.

    graphics/01fig10.gif

Example 1.4: Graphing a Horizontal Line

Graph the equation y = 3.

Solution

This one is a little tricky; at first glance it doesn't look like the equation of a line. However, if you think of it as

0x + 1y = 3

it matches the Ax + By = C form.

  1. Try to find three individual solutions that satisfy the equation.

  2. Pick three x values, such as 0, 1, and –1. When you plug them into the equation, you get the three ordered pairs (0,3), (4,3), and (–4,3). Actually, no matter which x values you choose, the corresponding y value is always 3.

  3. Plot the three points, and draw the line connecting them.

    The graph of the line y = 3 is shown in Figure 1.11.

    Figure 1.11. The complete solution for the linear equation y = 3.

    graphics/01fig11.gif

NOTE

At this point, we have defined the equation of a line in 2D: Ax + By = C. The next section investigates this linear equation in even more detail and extends it to 3D.


Self-Assessment

State whether the following equations are linear (that is, if the complete solution is a line):

1.

2x – y = 5

2.

–x + 5y = 0

3.

x = –1

4.

y + x2 = 5


Graph the following linear equations:

5.

x – 2y = 0

6.

–3x + y = 4

7.

x = 1


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