2. Revision: What is an Equation

The first step in solving problems using algebra is to first represent the given information as an equation. This step is very similar to translation from one language into another, say from Marathi to English. Suppose you want to translate the sentence ``Ram ate a mango'' into English; then you must determine what are the words in English that mean the same thing as the respective Marathi words, and then you have to combine those English words in a suitable manner. Writing an equation is similar; instead of translating from Marathi to English, you are translating Marathi into the language of mathematics, in which equations are like sentences! Once the given information is translated into equations, the next step is to manipulate the equations to get the answer we want. This is called solving an equation. In this chapter we will revise what you already know regarding writing and solving equations.

1 Marathi sentences to Equations

Consider the following sentences:

Sentence 1:

When 13 is multiplied by 4 and 20 added to the resulting number, the answer is 72.

Sentence 2:

When Raju's age is multiplied by 4 and 20 added to the resulting number, the answer is 72.

Can you write the above sentences using mathematical symbols instead of words? The first one should be very easy- you could write:

Sentence 1 in Mathematical form: $13\times 4+20=72$

What you have written down is an Arithmetic equation.

The second sentence looks more difficult, because we are not told Raju's age. But that is no problem in Algebra. We simply write the phrase ``Raju's age'' as if it were a number and keep going. So if we want to mean the number obtained by multiplying Raju's age multiplied by 4, we simply write ${\mbox{\rm {Raju's age}}}\times 4$. If we want to mean Raju's age multiplied by 4 and 20 added to it, we simply write ${\mbox{\rm {Raju's age}}}\times 4+20$. So in this manner we can translate sentence 2.

Sentence 2 in Mathematical form: ${\mbox{\rm {Raju's age}}}\times 4+20=72$

This in fact the central idea in algebra - even if we dont know the value of a certain quantity, we just pretend it is a number and write it down along with the other numbers that we know. What we have written down, is an algebraic equation.

1.0.1 Exercises:

Write the following sentences as equations (Click for solutions):

1. When 17 is multiplied by 9 and 3 subtracted from it the answer is 150.¹
2. When 2 is subtracted from 25 and the result multiplied by 4 the answer is 92.²
3. When Sanju's age is multiplied by 7 the answer is 91.³
4. When Sanju's marks in Algebra are divided by 6 the answer is 16.⁴
5. When Raju's weight is doubled and 5 added to it the answer is 55.⁵

1.1 From equations to Marathi

Suppose somebody wrote on a piece of paper `` $13\times 4+20=72$'', would you know what it meant? Could you write the meaning in Marathi, without using mathematical symbols? That should be easy- you would just write ``When 13 is multiplied by 4 and 20 added to it we get 72.''

Likewise suppose somebody wrote on a piece of paper ``Sanju's age $\times 4+20=72$''. How should you interprete it? You should say to yourself, ``This is an equation about Sanju's age. Sanju's age is not given explicitly, but whatever Sanju's age is, if it is multiplied by 4 and 20 added to it the result will be 72.'' So if you are to write the meaning of the equation without using mathematical symbols, you could write ``When Sanju's age is multiplied by 4 and 20 added to it we get 72.''

Exercises:

What do the following equations mean? Write the meaning in Marathi, without using mathematical symbols.

1. Distance between Mumbai and Pune$/40=4$
2. (Raju's height in inches-3)$/12=4$

1.2 How do mathematicians write equations

Mathematicians write equations slightly differently. First, mathematicians dont like to write long phrases or even complete words in equations. Instead, it is customary to say, ``Let us use the letter $x$ to denote Raju's age. In whatever follows if you see $x$ pretend that it means Raju's age.'' A mathematician would thus write sentence 2 from Section 2.1:

Mathematician's Sentence 2 Let $x$ denote Raju's age. Then 4 $\times x$ + 20 = 72

Mathematicians are in fact even more lazy - they commonly omit the ``$\times$'' symbol. So the equation above would be written as 4$x$+20=72. As another example, mathematicians would write ``4(3+9)'' rather than ``4$\times$(3+9)''. The $\times$ symbol is not always omitted. Suppose you wish to write 53$\times$39, then writing it as 53 29 would be confusing - someone might read it as the single number 5329 (Five thousand three hundred twenty nine). In this case, it is customary to not omit the $\times$. There is a further convention while making such short forms: if the product involves a number and a variable, then the number must come first, i.e. it is appropriate to write $4x$ when you mean the product of 4 and $x$, but not $x4$. This is only a convention, but you must follow it.

Exercises

Write the following sentences as equations as mathematicians would write them (Click for solutions):

1. When Sanju's age is multiplied by 7 the answer is 91.
2. When Sanju's marks in Algebra are divided by 6 the answer is 16.
3. When Raju's weight is doubled and 5 added to it the answer is 55.

The $\times$ operator has been omitted in the following expressions. Write the expressions showing where the $\times$ operator is implied.

1. $5x-7=43$
2. 4(37+98)
3. $7(x-9)+34(97-38)$

2 Solving Equations

Solving an equation means finding a value for the unknown variable such that the statement equivalent to the equation is true. Suppose we are given the equation $2x+5=13$. In Marathi, we read this equation as ``When the number $x$ is multiplied by 2 and 5 added to the result we get 13''. If instead of $x$ we wrote 5, the statement would read ``When the number 4 is multiplied by 2 and 5 added to the result we get 13''. This statement is true - and so we say that the equation has been solved and the value of the solution is $x=4$.

An equation is solved by deducing new equations from it. For making these deductions we just have to observe that both sides of the $=$ sign can really be thought of as just numbers. On the right hand side we actually have the number 13. The left hand side is $2x+5$ which doesnt look like a number immediately, but if we knew what number $x$ represented, $2x+5$ simply represents 5 added to the double of that number. With this observation there is a very simple rule for making deductions.

Fundamental idea in solving equations: Given any equation that we know to be true, we can get another true equation if we perform the same operation on both sides.

Let us use this rule on $2x+5=13$. One operation that we could perform is adding 1. So we could get $2x+5+1=13+1$. The left hand side should now be read as the number obtained by doubling $x$ (whatever value it represents) and then first adding 5, and then adding 1. But if we first add 5 to a number and then add 1, it is as good as adding 6. So we could write the left hand side as $2x+6$. The right hand side has $13+1$ but instead of this we could just write 14. So we have a new equation $2x+6=14$. We can be sure that the new equation is true if the first equation is true.

Of course, not all equations that we can deduce are equally important. Consider what happens when we subtract 5 from both sides of $2x+5=13$. Then the left hand side becomes $2x+5-5$, which represents the double of $x$ to which we first add 5 and then from which we subtract 5. But adding 5 followed by subtracting 5 is equivalent to doing nothing. Thus we can write the left hand side as just $2x$. The right hand side is 13-5, or in other words 8. Thus the new equation that we deduce is $2x=8$. This new equation is special in that it is simpler than the other equations. In this we only talk about doubling $x$, but both the equations $2x+5=13$ and $2x+6=14$ involve not only doubling $x$ but also adding something to the result.

Consider now what happens if we divide both sides of $2x=8$ by 2. The left hand side becomes $2x/2$ or in other words the number obtained by first doubling $x$ and then halving it. But since doubling any number first and then halving it leaves it unchanged, on the left hand we might simply write $x$. On the right hand side we have $8/2$, i.e. 4. So the new equation is just $x=4$. Translating this into Marathi we may write ``When nothing is done to number $x$ we get $4$''. In other words $x$ must be 4, i.e. 4 is the solution for $x$.

In general we can state the method for solving equations as follows:

Method for solving equations: Keep deducing new equations from the given equation such that equations become simpler, until you get an equation of the form $x=\ldots$, where upon you have the solution.

2.1 Another problem

Consider the following problem:

I have some number of pedhas with me. The number of pedhas that Raju has is twice the number of pedhas that I have. Together we have 30 pedhas. How many pedhas do I have?

In this problem there are two unknown quantities: the number of pedha's that I have, and the number that Raju has. So we could begin by saying ``Let $x$ denote the number of pedhas that I have and let $y$ denote the number of pedhas that Raju has.'' However, we dont need to worry about another unknown $y$, by reasoning as follows.

If $x$ denotes the number of pedhas I have, then the number of pedhas that Raju has must be the same as the number $x$ multiplied by 2. But we have decided to denote this number as $2x$. Further, the number of pedhas we have together is the number obtained by adding the pedhas I have to the number that Raju has. So we can denote the total number of pedhas as $x+2x$. But we know this number is 30. So we can write $x+2x=30$. This is the equation we need to solve to find the number of pedhas I have.

Solving the equation: Let us see how we can solve $x+2x=30$. First observe that $x+2x$ represents the number obtained by adding $x$ to its double. More intuitively, if we have $x$ pedhas in a bag and have two additional such bags, together we would have 3 bags, each with $x$ pedhas in it. So in other words we would have $3x$ pedhas totally. The general principle is that if you multiply $x$ by two numbers and take the sum, then it must be the same as first adding up the two numbers and then multiplying them by $x$. So in short, the equation we have can be written instead as $3x=30$.

But now we know how to solve this equation. Let us divide both sides by 3. Then on the left hand side we will have $3x/3$ which is the number obtained when $x$ is first multiplied by and then divided by 3, i.e. the number $x$ is effectively left unchanged. So the left hand side is just $x$. On the right hand side we have $30/3$ which is 10. So the new equation is $x=10$. Thus I had 10 pedhas with me.

3 Summary

Algebraic equations are just like sentences. Just as sentences in Marathi are meant for representing information, so are equations in algebra. Equations in algebra are meant to represent information about numbers. An algebraic equation consists of two numerical expressions, one on either side of an ``='' sign. The equation asserts that the number represented by each of the expressions must be the same. The expression may contain numbers as well as letters which stand for numbers that we dont know. The algebraic equation asserts that if the values represented by the letters are known and if they are written in place of the letters, then the expression on both sides of the ``='' sign would evaluate to the same number.

Since both sides of the equation (i.e. expressions on both sides of the ``='' sign) represent numbers, we can perform arithmetic operations on them. But to maintain equality of the numbers, we must perform the same operation on both sides. This is how we can generate new equations from any equation.

By performing properly chosen operations, we can simplify the form of the equation. Once we get to an equation in which only the unknown letter is on one side and a number on the other, we have solved the equation.