You will sometimes hear people speak of a “straight line” when explaining a problem or giving directions to someone else. Do lines bend? In the classical case of line, the answer is no, so saying “straight line” is redundant. An interesting question then arises: “What makes a line straight?” One argument is that the angle through which the line moves is constant. Imagine Descartes explaining this application of numbers in a plane to his contemporaries. He might have said that the motion of the line is constant—that is, for every change in the horizontal motion, the corresponding change in the vertical motion must be proportional. He might also have taken advantage of everyone's picture of a mountain to explain this principle. There is some debate among the historians of mathematics about whether such conversations really happened, but the story does help to explain that the symbol chosen for slope is the letter m. The slope of a line is the ratio of the vertical change in the line (the rise) over a corresponding horizontal change (the run). Using the Greek uppercase letter – (delta) to stand for “change in,” we can represent these two geometric interpretations by the formula
Example: Find the slope of the line joining: (a) point E(-4, −3) to point A(3, 2); (b) F(-2, −5) to B(2, 3); (c) E(-4, −3) to C(-2, 4); (d) M(0, −6) to H(4, −2)
In all four cases, the slopes of the lines are positive numbers. Use a ruler and a pencil to graph each of these lines. As you look at the graphs from left to right (which is the way all graphs in mathematics are analyzed), you should notice that all of the graphs are rising, or increasing.
Example: Find the slope of the line joining: (e) K(0, 6) to B(2, 3); (f) D(-5, 2) to L(-3, 0); (g) C(-2, 4) to G(2, −4); (h) A(3, 2) to J(5, 0)
The slopes of the lines are negative numbers in these four problems. Draw the lines and observe that they all fall, or decrease, as they are examined from left to right. Also note that two of the lines,
Two lines will be parallel whenever their slopes are the same. Equal rates of change guarantee that the lines will not intersect.
Example: Find the slope of the line joining: (i) D(-5, 2) to A(3, 2); (j) L(-3, 0) to J(5, 0); (k) F(-2, −5) to C(-2, 4); (l) G(2, −4) to B(2, 3)