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Linear Relationships and Proportions The equation = + represents a generic line with slope and y-intercept . This represents a linear relationship between variables and . A special case is when = 0. In this case, the linear relationship is now = , and this special case is called a proportion between variables x and y. We can summarize: • • Not all linear relationships are proportions. If the line does not pass through the origin (equivalently, if ≠ 0), then the linear relationship is not a proportion. All linear proportions are linear relationships. Examples: a) Suppose you are told that 1 mile is equal to 1.62 kilometers. Using m for miles and k for kilometers, we could write = . This is a proportion. You can tell because 0 miles is the same as 0 kilometers. The equation could also . be written as = 1.62 or = . or = 0.617. All of these are equivalent, and all represent the same proportion. If you were to graph k on the y-axis, and m on the x-axis, you’d have a line passing through the origin with slope 1.62. The units of this slope are . This means that every one mile corresponds to 1.62 kilometers. b) The linear relationship between degrees Fahrenheit are degrees Celsius is given by the equation = + 32. However, this is not a proportion. It does not go through the origin (for example, 0 degrees C is the same as 32 degrees F). c) If I drive my vehicle at 40 miles per hour, I could set up a linear relationship " = 40$, where d is distance in miles and t is time in hours. This is a proportion since at time = 0 (i.e. I haven’t yet started driving), I have gone 0 miles. The slope is 40, and is read as 40 . %& d) If I drive my vehicle at 40 miles per hour, but start 20 miles from home, then the linear relationship between d, my total distance from home, and t, my time in hours, is given by " = 40$ + 20. In this case, the variable quantities d and t are not proportional. (Why?). Parts (c) and (d) are similar. The first is a proportion, the second is not. A lot depends on exactly how we define the meaning of each variable. You’ll note in part (d), the distance variable includes a value of 20 representing the starting distance from home. Change in a Variable ' Let x and y represent two generic variables. If x and y are proportional to one another, then we can write ( = , which is the same as = . The constant k is called the constant of proportionality and is seen as the slope of the graph of the line = . As x changes, so does y. In fact, we can relate the change in y, written ∆ with the change in x, written ∆, as ∆' a proportion, and it will be the same value as k. That is, = also. ∆( What’s interesting is the case when x and y are linearly related, but not proportional. In such a case, the ratio of the rates of ∆' change in y and in x are proportion also! In other words, if = + , then ∆( = . In the example above, parts (b) and (d) were examples in which the variables were not proportional, but their changes are. ∆* In example (b), the variables F and C are not proportional, but ∆+ = , so ∆ and ∆ are proportional. A change of 5 ∆, degrees Celsius translates to a change of 9 degrees Fahrenheit. In part (d), we could say that ∆- = 40. That means ∆" and ∆$ are proportional (but d and t are not). A one hour change in time corresponds to a 40 mile change in distance, regardless if I start at home, 20 miles from home, or 500 miles from home. Be aware of this subtlety in language! Read the questions carefully. We will see this concept re-appear as we develop the basis for calculus.