### Very Short Introduction to Linear Equation Of Two Variables(Top E-Learning app)

(Top E-Learning app)

Hey there everyone! How are you all?

The answer can be good or bad, or maybe just nothing. But as long as it is positive and hopeful, things will go uphill.

Now, with this optimism let’s head into learning something new, for those who haven’t studied this before, and brushing up on old concepts, for those who have studied it before. (Top E-Learning app)

Let’s just maybe get a little nostalgic.

The good old pre-Covid days, when schools used to be open! Good times, eh? Yeah, so tell me how have your parents been about what you spend during your school hours? Did you get pocket money?

Did you not get pocket money? Was it on a daily basis? Was it on a weekly or monthly basis?

Also, some of you must be reminiscing of the days when you would leave school and the Ice-cream kaku would be right there to give you your favourite one. Or maybe the kaku who sold gola, already had tons of students flocking around him.

So, what if one day you have, say Rs. 30 of your pocket money to spare (considering you get some! I didn’t , so I know how you feel.), and you see, there are two kakus there that day.

The phuchhka Kaku and the Ice cream kaku. Now you being the hungry kid, can’t figure out which one to have.

It’s a hot summer day, so an Ice-cream wouldn’t hurt, but the craving for those phuchhkas is ever undying.

Now, you might want to ask the price of each ice cream and that of the portion of phuchhkas. So let’s say, Rs. 10 can fetch you one ice cream and Rs.5 can fetch you 3 phuchhkas (one portion).

Now, what do you do. You think, “What if I buy only ice creams? How many will I be able to buy?” So you do your division in your head, and arrive at 3 . (30/10). You do the same thing for the phuchhkas. How many do you get? 6? (30/5).

But no, you get 6×3=18! Each portion has 3 pieces remember?

But, is there a way you could get the best of both worlds? Enjoy a little bit of this and a little bit of that. Let’s see.
Let us take the number of Ice-creams to be I and number of phuchhka portions to be P.

Now, we know the price of each I to be Rs.10 and that of each P to be Rs.5 . So the money you spend on I could be given by Rs.10xI and for P, it will be Rs.5xP.

And what is the maximum amount you could spend? Rs.30. Can’t spend more than what you have, right?
So , could there be a simpler way could represent all this? Yes!
10 x I + 5 x P = 30

A very important understanding here is, you are NOT SAVING ANY MONEY, YOU’RE SPENDING ALL OF IT. Which is natural, so don’t worry!
Back in school, even I did not know how to save.

Whatever little I had, I would make it a point to spend all of it on things like Chips and toffees.

Now, this expression,
10 x I + 5 x P = 30

Is what we call an equation. And it has two things in it that we need to find. The value(s) of I and that(those) of P. I and P, here, are called variables. This means that they could have different values.

Changing the value of one would change the value of other.

But right now, neither of the values are known to us, hence they are unknown variable. Now let’s go on to understand what all this actually means –
Now, what if, we take I to be 1, ie, say you definitely want to have 1 ice cream? Then how many P’s can you have?
10 x 1 + 5 x P = 30
10 + 5 x P = 30
5 x P = 30 – 10
5 x P = 20
P = 20/4
P = 4

So, we see that when we when we assign some value to I, that is, make it an known variable, P, the unknown variable, gets a value.
Now, what if, you want to have an ice cream and also treat your friend to one, then I will become 2.

Then the value of P can be calculated as follows-
10 x 2 + 5 x P = 30
20 + 5 x P = 30
5 x P = 10
P = 10/5
P = 2
So, here we see that P will be 2.
So you see, how the value P keeps varying with respect to each value of I. This is why they are called variables.

Also, you must recall, that in the very first instance, the first calculation you did was to find out how many of each you would be able to have if you buy just one of them. Then I was found to be 3, when P was 0; and P was found to be 6, when I was 0.

So how many sets of values do we have for each of I and P now –
10 x I + 5 x P = 30

therefore,

(I, P)

(i) (0, 6)

(ii) (1, 4)

(iii) (2, 2)

(iv) (3,0)

So, once we do this, we see that, not only can you have both the things , but also have a choice ( i , ii) in what combination you want to have both the things.
Now, I know most of you must be thinking what would have happened if I had decided to have 1 portion of phuchhkas, i.e, my P would have been 1.
Let’s see.
That would have lead us to write equation in the following form –
10 x I + 5 x P = 30
10 x I + 5 x 1 = 30
10 x I + 5 = 30
10 x I = 30 – 5
10 x I = 25

I=25/10

I=5/2= 2(1/2)

Now, you tell me, is it possible to buy 2 and a half ice creams? Well of course, you can eat 2 and a half ice creams, if your friend id generous and wants to share half of their ice cream with you even after you have had two of yours!

But, I am sure the shopkeeper wouldn’t like selling half an extra ice cream right?
So , here we see, that even for the values that we get the for the Unknown variables. They should be INTEGERS.

Now, let us quickly recall our integers, once.
………….,-4,-3,-2,-1,0,+1,+2,+3,+4,…………………. and so on, right?
Now, what if you wanted to have, say more than 3, say 4 ice creams. Then what would have happened?
10 x I + 5 x P = 30
10 x 4 + 5 x P = 30
40 + 5 x P = 30
5 x P = 30 – 40
5 x P = -10
P = – 10/5
P = – 2

Confused?

This is why we should not think about spending more than we have!
Deciding to eat 4 ice creams, would cost you Rs.10 x 4 = Rs. 40. But do you even have Rs. 40?

So from here, we also see, that in the current scenario, we would need the values to the unknown variables to be not just INTEGERS, but POSITIVE INTEGERS.

But the most crucial thing to understand here is that, the two unknown variables need not necessarily always be positive integers. Mathematically speaking, they can take all real values from -∞ to +∞. This is the truth.

But in our case, we were dealing with a real life example.

In the vast big world of Maths and co-ordinate geometry(which we will see very soon in one of the next few blogs), -2 and 2(1/2), are very valid values among a sea of a lot of other values.

So kids, I hope you have understood, what all I have tried to say until now, and I am sure, I have been able to clear some basics around linear equations with two variables, based on their real life applications.