Eduki

Equations are also used with two variables to show how two quantities depend on each other. For example, the bill of electricity is dependent on the units of electricity consumed. The traffic in the street is dependent on the time of the day etc.

The relation may not be as straightforward all the time but they can certainly be equated with an equal sign.  For example, if a pen cost 2 dollars, two will cost 4 dollars, 3 will cost 6, and so on. No matter how big the number of pens will be, the price will exactly be twice that. This perfectly represents a situation where we don't have to pinpoint an exact number of pens and their corresponding price. Rather, we can choose a placeholder variable for both and relate them with an equation. 

price of pens = 2 𝑥 (number of pens)

If the number of pens is denoted by 𝑥, their price is denoted by y, then the relationship would be

y=2𝑥

The concept of greater and smaller

Let's suppose another situation where in a university there is one professor each for 20 students. Then, which of the following would be correct?

No. of students = 20 x (No. of professors)

Or No. of professors = 20 x (No. of students) 

The easier way of finding the right answer is to understand which is greater, the number of students or professors. If there is one professor for 20 students, two professors for 40 students, etc. It can be seen that students are always more in number than professors. 

 

If the larger number (the number of students) is multiplied by 20, then it will become even larger and the quantity can never be equal to the smaller number (number of professors). Thus the smaller number is multiplied by 20, then it will become equal to the larger number. Thus,

No. of students = 20 x (No. of professors)

If f represents the number of professors and d represents the number of students then 

d=20 x f

The same equation can be written as f=d/20, if we divide both sides of the equation by 20.

In the two representations, the variable that is written on the right-hand side with some operation being performed on it is the independent variable whereas the variable written on the left is a dependent variable. In f=d/20, f is dependent and d is independent. Whereas in d=20f, f is independent and d is dependent. How the equation is written doesn't have much effect on the actual relationship between the two quantities. 

In both the examples above, it is seen that one quantity can be found by multiplying another quantity with some factor. It also implies when one quantity doubles, the other quantity is also doubled. It applies to any number acting as a factor. These types of relations are known as proportional relationships. All relationships do not follow the proportionality property. For example, 

The height of a rectangular window is 6 inches less than the width. Let w represent the width of the window and h be the height. 

We know that 6 is the difference between height and width in which the width is larger so either 6 has to be subtracted from the width or 6 has to be added to the smaller quantity that is the height. 

w-6=h

Or w=h+6 

Here when w=1, we get h=1+6=7

When w=2, we get h=2+6=8

Here when the value of w is doubled from 1 to 2, the value of h doesn't double from 7 to 14, instead it becomes 8. So the relation is not proportional.