What does it mean to say that a line is straight?

Describe "straightness" in as many different ways as you can.

So far, we have algebraic descriptions of lines, segments, and rays only if they are parallel to one of the coordinate axes. But most lines are not like that; what about them? How can we use algebra to describe these other lines, segments, and rays?

An ant is at (0, 0), the origin of a coordinate plane. It wants to walk to (10, 5) by the shortest path, which is along a straight line. Attached to one ankle it has an ant-sized electronic gadget that tells it the exact coordinates of each point it steps on. How can it use this gadget to stay on the straight path from (0, 0) to (10, 5)?

1. On a piece of graph paper, draw a pair of coordinate axes. Mark the point (10, 5).

2. Use a ruler to draw a straight path from (0, 0) to (10, 5).

3. Draw three other paths from (0, 0) to (10, 5), any way you want.

4. Mark four points on each path you drew. Use the coordinates of these points to fill in the four small tables in Display 3.14. (As an example, one point of the straight path, (2, 1), has been picked for you and filled in.)

5. Do any of the four tables have an obvious pattern? If so, describe the pattern(s).

6. What advice would you give to the ant?

Learning Outcomes
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After studying this section, you will be able to:

Describe what it means to say that a line in the coordinate plane is straight;

Find the slope of a line from two of its points;

Draw lines described by equations of the form y = mx;

Determine from an equation whether or not a point is on a given line through the origin.