Linear Relationships

Let's return to our list of equations.

1. y = 4x + 3
2. y = 4x^1/2 + 3
3. y = 4x - 2z +3
4. y = 10/x
5. y = 1.1^x
6. lny = 4lnx
7. y = 3 + 4x²
8. lnQ = -2lnP +1.5lnY

The first equation is by far the simplest. We can begin by building a table to represent the relationship.  In the first column we put the numbers 1 through 40 and the next column we put in a number 4 times the number in column 1 plus 3.  This would produce the table below.   

The Table

This table signifies a relationship between x and y such that every time x increases by 1, y increases by 4. If we were looking at a graph, the relationship between changes in x and changes in y show up in the slope which would be 4.  This is the distinctive feature of linear relationships- their graphs are straight lines.

The Graph

In the equation form, the slope (rate of change) appears as the coefficient of x. The other parameter in the equation is 3 which tells us that when x = 0, y is equal to 3. In a graph, this would be the y intercept.  When x = zero, then y equals 3.

The Equation

y = 4x + 3

If we compare equations 1 and 7 we find there is only one difference. In equation 1 the variables y and x are raised to the first power (1), while in equation 7 x is raised to the second power (2). This is a significant difference. At this point you should be able to recognize equations where the variables are raised only to the first power.  These are called linear equations and possess certain desirable properties. The most obvious property is the fact that the rate at which y changes for any change in x is independent of the value of x. In the linear equation, if we are at x = 1 and move to x = 2, y increases by 4. The rate of change Dy/Dx = (11-7)/(2-1) = 4. If we begin at x = 30, however, and move to x = 31, the rate of change (Dy/Dx) = (127-123)/(31-30) = 4/1 = 4.

One can conclude from this simple example that when you have a linear equation the rate of change is a constant.  In the above example, every time x increases by one unit, y increases by 4.  We can generalize for the generic equation that you should recall from high school.

y = mx + b

In this generic example, the slope equals the rate of change.

(1)  Dy/Dx =a.   

The rate at which x causes y to change is equal to the coefficient of x. As we will see many times in this section, we can interpret this relationship as one equation with three unknowns. Using the 'laws' of algebra, we can rewrite this equation in the following two ways - each providing us with a different variable on the left hand side of the = sign.

(2) Dy = Dx*a

(3) Dx = Dy/a

The first of these equations can be used to answer questions such as; how much will y increase if x increases by 3? if x decreases by 4, they what will happen to y? In the case where the coefficient was 4, x would increase by 12 when y increased by 3. If on the other hand you happened to know that y increased by 12, you would know x must have increased by 3. You could answer this easily by referring to the second form of the equation.

Before we leave our discussion of linear equations, let's return to the list of equations and see if there are any other linear equations. The secret is the exponent for the variables must be equal to 1. The fourth equation looks promising, but you will note that X is in the denominator so we could rewrite it as y = 10*X^-1 so its exponent is really -1. Equation 3, meanwhile, does satisfy our condition for a linear relationship, the only difference is that y depends on two exogenous variables, x and z. The nature of the relationship is identical to that which we discussed for the first equation. In the linear equation, if we are at x = 1 and z = 2 and move to x = 2, then y increases by 4. The rate of change Dy/Dx = (7-3)/(2-1) = 4. If we begin at x = 30, however, and move to x = 31, the rate of change (Dy/Dx) = (123-119)/(31-30) = 4/1 = 4. Once again the coefficient of the variable gives us the slope of a graph and the rate of change between y and x.

Similarly, the coefficient of z indicates the rate at which y changes when z changes. In equation 6, y will change 2 for every one unit change in z. In fact, you can see this in the table below. You note that as you move across one row you are comparing the values of y, for a given value of x, as z changes from 0 to 2. For example, if x = 3 and z = 0, then y = 15. If you hold x constant at 3 and allow z to increase to 2, the value of y decreases to 11. In this situation Dy/Dz = (11-15)/(2-0) = -4/2 = -2, which is the coefficient of z.

Now let's move on to a discussion of nonlinear relationships.