The relationship between Fahrenheit (temperature in America) and Celsius (temperature as measured and reported on the rest of the planet) is linear. If we let F = Fahrenheit and C = Celsius the classical ways to express these relationships are C = (5/9)(F - 32) and F = (9/5)C + 32. The new way to do it is: add 40, multiply, subtract 40 -- it works both ways. To put it in mathematical terms: F = (9/5)(C + 40) - 40 and C = (5/9)(F + 40) - 40. The new, simpler way, which is both easy to remember, and so easy to calculate that people can do the calculations mentally, is only possible because the equations are linear and therefore susceptible to such manipulations.
Elevators always have the maximum allowable weights posted. These maximums assume that the average person weighs 150 pounds. If any equipment is riding in the elevator, the actual weight (W) is computed with a linear equation. W = 150P + E where P is the number of people and E is the weight of any equipment that is riding up in the elevator. Sometimes when equations are supposed to be below a certain level they are written as linear inequalities. If we want to accurately express the condition when the elevator is not overloaded we should write 150P + E < MAX where MAX is the posted limit.
The advantage of linear equations with one variable is that they are easy to manipulate. The real power of linear equations can be seen when there is more than one variable. Graphing two linear equations and looking at where the lines intersect can tell you the values that fit in both equations. Consider, for example, comparing two cell phone plans. Plan A costs $25 a month + 5 cents a minute. Plan B costs $10 a month plus 10 cents a minute. If you have a good idea how many minutes a month you will use, which plan is best? If we graph both equations (A = 25.00 + 0.05M and B = 10.00 + 0.10M) they intersect at the points where M (Minutes) is 300. For less than 300 minutes line A is above line B. If you use less than 300 minutes a month, plan B is better.