Linear Equations

Let's go back to the relationship between advertising and sales, as shown in Table A.1. What if you need to know how much sales the firm could expect if it spent \$5,000 on advertising next month? What if it spent \$8,000, or \$9,000? Wouldn't it be nice to be able to an swer questions like this without having to pull out tables and graphs to do it? As it turns out, anytime the relationship you are studying has a straight-line graph, it is easy to figure out the equation for the entire relationship. You then can use the equation to answer any such question that might be put to you.

All straight lines have the same general form. If Y stands for the variable on the vertical axis and X for the variable on the horizontal axis, every straight line has an equation of the form

Y = a + bX, where a stands for some number and b for another number. The number a is called the vertical intercept, because it marks the point where the graph of this equation hits (intercepts) the vertical axis; this occurs when X takes the value zero. (If you plug X = 0 into the equation, you will see that, indeed, Y = a.) The number b is the slope of the line, telling us how much Y will Weeks Since Launch change every time X changes by one unit. To confirm this, note that as X increases from 0 to 1, Y goes from a to a + b. The number b is therefore the change in Y corresponding to a one-unit change in X—exactly what the slope of the graph should tell us.

More generally, if X changes from some value X1 to some other value X2, Y will change from

AY = Y2 - Y1 = (a + bX2) - (a + bX1) = a + bX2 - a - bX1

Dividing both sides of the equation AY = bAX by AX, we get

If we subtract Yj from Y2 to compute how much Y has changed (AY ), we find that confirming that b really does measure the slope.

If b is a positive number, a one-unit increase in X causes Y to increase by b units, so the graph of our line would slope upward, as illustrated by the red line in panel (a) of Figure A.3. If b is a negative number, then a one-unit increase in X will cause Y to decrease by b units, so