MATHS - 5.EQUATIONS AND IDENTITIES

EQUATIONS AND IDENTITIES is a basic concept introduced in middle school but useful with application in higher studies too.

Many are not clear of the basic difference between the two terms and the power of proper understanding to solve even complex problems.. When we ask SOLVE the following …the reactions vary very interestingly.. giving an idea of their psychology too…

1. When we ask what is x if x+2=5
   Many respond by saying x = 3..This is a first degree equation in x with one solution for x .
2. When we ask what is x if x^2 =4....…Many respond by saying …x = +2 or –2 …This is a second degree equation in x with 2 solutions for x .
3. When we ask,solve x+y = 5.....every one goes silent for some time…When prodded.. one says…we can not solve.another says the problem is wrong. another says x = y - 5….When asked ..is x = 2 and y = 3 a solution …they pick up the thread and say … x = 1 and y = 4 is also a solution ..x = 3 and y = 2 is also…etc…now they agree that it can be solved and there are infinite solutions.

YES correct but does infinity mean every thing ? No….infinite .. means many many many …..but not all or any value for x and y

4. Then we go to ..solve...(x+y)^2=x^2+y^2+2xy...with the confidence gained from above problem ,students respond by giving different solutions for x and y . .. once again they say there are infinite solutions. But now they agree this infinity is different ...it means all values or any value of x and y is a solution. Then this is called an IDENTITY…it is true for any or all values of x and y.
5. The examples 1,2 & 3 are all equations ..they are all solved obtaining one or many(but not all or any value ) or some times no value for x and y. They are true only for those values of x and y. That is the difference between equation and identity. Number of solutions depend on the nature of the equations. It is taught in high schools that example 1 is a first degree equation in x and has one solution, the example 2 is a second degree equation in x and has 2 solutions and in general there will be as many solutions as the degree of the equation.
6. Now let us see the power of this simple concept of high school level to solve a problem at a higher level
   
7. Show that

[(x-a)(x-b)/(c-a)(c-b)]+[(x-b)(x-c)/(a-b)(a-c)]+[(x-c)(x-a)/(b-c)(b-a)]=1

8. Looks complicated. But it does not need even a pen and a paper. It is really an oral

problem. Let us use our concepts.. If we put x = a we find it tallies, that is left

hand side is equal to 0 + 1 + 0 = 1. Similarly it tallies for x=b and x=c. But it is

a second degree equation in x and so can have only 2 solutions for x… Still we

find a third solution. So it must be an identity. So it is true for all values of x . PROVED.

So when you find such a problem in your school exam, don’t say it

is out side syllabus. At best you may say , you were not shown the light .

TAIL PIECE : Let us encourage the spontaneous student responses mentioned in

the above examples. Some times we may try to guide them to better

responses if needed. For , if every one at that age is to give a

measured response , then it will be a very dull life and world. Some

may recall some news readers in TV earlier…maintaining the same

face to announce the death of a National Leader or the victory of

a National Team in a world class event.