MATHEMATICS > ALGEBRA > GEOMETRY

SUM OR DIFFERENCE OF TWO QUANTITIES

For similar terms, it is not necessary that sign and coefficiet of each variable should also be equal. One can think of an algebric term to be part of coefficient part and variable part. Coefficient part contains sign and number ( number may be whole or in fraction). Variable part contains one or more variables only. These variables may also have powers. Then condition of similarity in between two terms is that variable part of both terms should exactly be the same. Every variable of first term should also be variable in second term and powere of each variable in each term should be the same.

Both terms are similar terms because both terms have only one variable which is x. Both terms have only one variable In both terms power of variable x is one. Variable part of both terms, which is x, is same.

Terms are not similar because variable part of terms of first term is x² and variable part of second term is x³. Here powers of variable x are different.

Both terms are similar. Both term contain 3 variables x, y and z. Power of variable x in first term is 1 and power of variable x in second term is also 1.Simlarly, power of variale y and z are also equal in both terms. Thus we see that variable part of both terms is equal. Only coefficients differ.

Both terms are not similar. In each term there are three variables x, y & z. Thus total number of variables is same for both terms. If we examine powers of each variable, we see power of x in first term is 1 whereas power of x in second variable is 2. Similarly, power of y in first varial is 2 whereas in second term power of y is 1. Power of z in both terms is one. Thus we see that powers of variables are not same in each term. Variable part of first varial xy²z is not equal variable part of second variable x²yz.

For converting sum of two similar terms into single term first we notice the sign of each term. If both terms have same sign of + or same sign of -, them we add their coefficients and we write the resultant term as under:

First we put sign of term 9which is sign of given terms) and then sum of coefficients and then variable part.

Solution: Here we see that both terms are similar terms because their variable part xy is exactly the same. Again we see that both terms are of + sign [where there is no sign before a term, a + sign is understood]. Therefore, we have to add the coeffficients. Now we can write the resultant term as + 10 xy. Being single first term and sign being +, we can leave + sign. Hence term is 10 xy.

Solution: Here we see that both terms are similar terms because their variable part xy² in both terms is the same. Again we see that both terms are of - sign. Therefore, we have to add the coeffficients. Now we can write the resultant term as - 4 xy².

Now, variable part of both terms is same as xy². Therefore, both terms will give us a single term. On adding these terms we will get:

WHEN TWO SIMILAR TERMS ARE OF OPPOSITE SIGN (ONE + AND OTHER -) THEN THEIR SUM WILL BE-

(SIGN OF THE TERM HAVING GREATER COEFFICIENT) (DIFFERNECE OF COEFFICIENTS) (VARIABLE PART OF TERM).

When two similar terms are of opposite sign and their coefficients are similar, their sum will be zero. Sign will have no effect because difference of coefficients will be zero and anything multiplied by 0 gives 0.

IF TWO TERMS ARE NOT SIMILAR, IT DOES NOT MEAN THAT THEY CANNOT BE ADDED. ON ADDITION THEY WILL GIVE AN EXPRESSION CONSISTING TWO TERMS.

Since variable part of first term is xy and variable part of second term is xy². Hence variable parts are different and therefore, terms are not similar. Therefore, they will not give a single term. Therefore, their sum will be written as such.

Since variable parts of both terms are different, therefore they cannot be converted into a single term. Hence answer is -3xy² +9xy. The answer can be written as 9xy -3xy². [it is customery to write first the term which has + sign but there is nothing wrong even if term having - sign is written first]

Sum of given terms = firest term + (second term) + (third term + (fourth term) + (fifth term)

Except first term, we put each term in small bracket and in between them we put + sign and then simplify the resultant expression by adding similar terms, if any.

Sum of given terms = firest term + (second term) + (third term + (fourth term) + (fifth term)

Sum of given terms= -9xy² + 3xy + 4xy² + x²y - xyz [except fifth term -xyz, all other tems within brackets have + sign. Therefore, multiplication of sign before the bracket with sign of term within the bracket has given + sign, whereas in case of fifth term, multiplication of + sign before the bracket with - sign of fifth term within the bracket has given - sign]

Now, variable parts of first and third term are same as xy². Therefore, on simplification, they will give single term. All other terms have different variable parts.

Or sum of given terms= x² + -5 xy²+3xy + xyz [it is customary to put positive term first]

Resultant expression cannot be simplified further. Hence answer is x² + -5 xy²+3xy + xyz

After putting both expressions in the above equation, we first remove bracket [for removing bracket, sign of each term within the bracket is multiplied by the sign appearing before bracket (here +)] and thereafter, we simplify the resultant expression by converting similar terms, if any, into a single term.

Now we see that first and fourth terms have same variable part x² and second and fifth also have same variable part xy. Therefore on converting similar terms into single term, we get:

sum of two given expressions =4x² + y² + y³[ x² + 3x² have given 4x² and +2xy & - 2xy have given 0 xy or 0]

IN THE SIMILAR MANNER WE CAN ADD ANY NUMBER OF EXPRESSIONS TO GIVE A SINGLE EXPRESSION.

Solution: Here first term is 9xy because from 9xy, the other term -3xy is to be subtracted. Term -3xy is to be subtracted. Therefore, -3xy is the second term.

For removing the small bracket, we will have to multiply sign of term within the bracket with sign before the bracket. Here, sign of term within the bracket is - and sign before the bracket is also -. Therefore, their multiplication will give a+ sign.

Now both terms are similar terms because their variable part xy is the same. Also both terms carry the same + sign, therefore,-

Now, Difference =first term - (second term) [do not forget to put second term within small bracket]

Difference = 7x² - 3x² [+ sign of 3x² on multiplication with - sign before the bracket has given - sign]

Variable parts of both terms are same as x<sup>2</sup>, therefore, they can be converted into single term. Therefore,

Difference=9x² - xy + 9 - x² - 3x -7 [Notice change of sign of each term within the bracket]

Now, first and fourth terms have same variable part, therefore they can be converted into single term. Also 9 and 7 are numbers and numbers can be added or subtracted independently if they are not coefficients to any variable, as may be required. Numbers used as coefficients can only be added or subtracted when their variable pats are the same.

While adding any number of terms, except first term all other terms are to be put in small brackets and a + sign is to be put in between them like Sum = first term + (second term) + (third term + (fourth term) + (fifth term)

While subtracting second term from first term, second term is always to be put in small bracket and a - sign is to be put in beween them like Difference =first term - (second term). Second term is the term which is to be subtracted.

While removing bracket, sign of term within the bracket gets multiplied by the sign immediately outside the bracket like -(3x) is -3x. Another example: (3x + 7y) is 3x +7y. In this case sign immediately outside bracket is +.

Two terms are said to be similar if their variable parts are same. For comparing variable parts of two terms, their complete part consisting of alphabets (variables) is to be taken together. They should have same number of alphabets (variables) and every variable should have same power. For example 3xy and 7xy are similar terms whereas 3x and 7xy are not similar terms. In this case, first term has variable part x & second term has variable part xy. Variable part of first term has one alphabet x as variable whereas variable part of second term has two alphabets x & y. Similarly, 3x²y & 3xy² are not similar terms. Their variable parts are x²y and xy². Though number of variable in both variable parts is 2 (first variable x & second variable y) yet their powers are different. Power of x in first term is 2 whereas in second term its power is 1. Similarly, in case of y, power of y in first term is 1 whereas in second term is 2. Therefore, these two terms x²y and xy² are not similar.

While comparing power in variable parts of two terms, powers of each variable in first term are to be compared with powers of same variable separately in second term. For example, if variable parts of two terms 8x³yz & 9xy are to be comapred, then powers x in first term will be compared with power of x in second term, power of y in first term is to be compared with power of y in second term and power z in first term is to be compared with power of z in second term. Two terms will be similar only if (i) number of variable in both terms is same; and (ii) powers of each variable in both terms are same. Both conditions should be satisfied. If any condition is not satisfied, terms are not similar. In this case both conditions are not satisfied. Hence terms 8x³yz & 9xy are not similar.

Numbers which are not associated with any variable (which are not coefficient of any term) may be added or subtracted (as may be required) to give a single number.

If any term has variable part only (does not have any number as its coefficient) then its coefficient is 1.

Any term, as a whole, which does not have a - sign is positive in nature and in coputations its' sign is taken as +.

Above all principles also apply in case of addition and subtraction of expressions.

SUM OR DIFFERENCE OF TWO QUANTITIES
A quantity may be a number or variable or variable with coefficient or combination (product) of several variables with or without coefficient or a term or an expression having single term or several terms. We can add two or more quantities and we can also subtract one quantity from another quantity to give a single term or expression. When we add two numbers, we always found a single number but in Algebra when we add two terms, we cannot get a single term unless both terms (where two terms are to be added) are similar terms. Similarly we cannot get a single term on addition of two expressions unless every term of both expressions is similar. By similar terms we mean that each term should have-
(i)	same variables;
(ii)	same number of variables; and
(iii)	every variable has same power.
For similar terms, it is not necessary that sign and coefficiet of each variable should also be equal. One can think of an algebric term to be part of coefficient part and variable part. Coefficient part contains sign and number ( number may be whole or in fraction). Variable part contains one or more variables only. These variables may also have powers. Then condition of similarity in between two terms is that variable part of both terms should exactly be the same. Every variable of first term should also be variable in second term and powere of each variable in each term should be the same.
Examples of similar terms and and terms which are not similar terms:
First term	Second term	comments
3x	5x	Both terms are similar terms because both terms have only one variable which is x. Both terms have only one variable In both terms power of variable x is one. Variable part of both terms, which is x, is same.
-2x	9x	Both terms are similar terms because both terms have only one variable which is x. Both terms have only one variable In both terms power of variable x is one.Variable part of both terms is equal and same.
3x²	2x³	Terms are not similar because variable part of terms of first term is x² and variable part of second term is x³. Here powers of variable x are different.
3xyz	5xyz	Both terms are similar. Both term contain 3 variables x, y and z. Power of variable x in first term is 1 and power of variable x in second term is also 1.Simlarly, power of variale y and z are also equal in both terms. Thus we see that variable part of both terms is equal. Only coefficients differ.
3xy²z	5x²yz	Both terms are not similar. In each term there are three variables x, y & z. Thus total number of variables is same for both terms. If we examine powers of each variable, we see power of x in first term is 1 whereas power of x in second variable is 2. Similarly, power of y in first varial is 2 whereas in second term power of y is 1. Power of z in both terms is one. Thus we see that powers of variables are not same in each term. Variable part of first varial xy²z is not equal variable part of second variable x²yz.

CONVERSION OF SUM OF SIMILAR TERMS IN A SINGLE TERM
For converting sum of two similar terms into single term first we notice the sign of each term. If both terms have same sign of + or same sign of -, them we add their coefficients and we write the resultant term as under:
First we put sign of term 9which is sign of given terms) and then sum of coefficients and then variable part.
Example: Convert 3xy + 7xy into single term.
Solution: Here we see that both terms are similar terms because their variable part xy is exactly the same. Again we see that both terms are of + sign [where there is no sign before a term, a + sign is understood]. Therefore, we have to add the coeffficients. Now we can write the resultant term as + 10 xy. Being single first term and sign being +, we can leave + sign. Hence term is 10 xy.
Example: Convert -3xy² -xy² into one term.
Solution: Here we see that both terms are similar terms because their variable part xy² in both terms is the same. Again we see that both terms are of - sign. Therefore, we have to add the coeffficients. Now we can write the resultant term as - 4 xy².

EXAMPLES OF ADDITION:
1. Find sum of 2x and 5x.
Solution: sum of both terms= first term + second term
Hence, sum of both given terms =2x + (3x) [In case of addition, it is not necessary to put term 3x within small bracket but I prefer this because in case of subtraction, it will be necessary. So I imphasize on putting expressions, to be added or subtracted, within a small bracket.]
Or sum of both terms =2x + 3x
Both terms are similar. Therefore, they can be added to give single term. Therefore,
sum of both terms = + 5x
Or sum of both terms=5x
Example 2. Find sum of 3xy² and -3xy²
Solution: sum of both terms= first term + second term
Hence, sum of both given terms = 3xy² + (-3xy²)
Or sum of both terms = 3xy² -3xy² [while removing small bracket we had to multiply + sign with - sign within the bracket which gave us a - sign. Therefore, second term has been put as -3xy².]
Now, variable part of both terms is same as xy². Therefore, both terms will give us a single term. On adding these terms we will get:
sum of both terms = 0
WHEN TWO SIMILAR TERMS ARE OF OPPOSITE SIGN (ONE + AND OTHER -) THEN THEIR SUM WILL BE-
(SIGN OF THE TERM HAVING GREATER COEFFICIENT) (DIFFERNECE OF COEFFICIENTS) (VARIABLE PART OF TERM).
When two similar terms are of opposite sign and their coefficients are similar, their sum will be zero. Sign will have no effect because difference of coefficients will be zero and anything multiplied by 0 gives 0.
IF TWO TERMS ARE NOT SIMILAR, IT DOES NOT MEAN THAT THEY CANNOT BE ADDED. ON ADDITION THEY WILL GIVE AN EXPRESSION CONSISTING TWO TERMS.
Problem: Find sum of 3xy and 5xy²
Solution:
Sum of both both terms = first term + (second term)
Here sum of both given terms = 3xy + (5xy²)
Or sum of both terms =3xy + 5xy²
Since variable part of first term is xy and variable part of second term is xy². Hence variable parts are different and therefore, terms are not similar. Therefore, they will not give a single term. Therefore, their sum will be written as such.
Sum of both terms =3xy + 5xy²
Problem: Find sum of -3xy² and 9xy
Solution: sum of two terms = first term + (second term)
Therefore, sum of given terms= -3xy² + (9xy)
Or sum of both terms= -3xy² + 9xy
Since variable parts of both terms are different, therefore they cannot be converted into a single term. Hence answer is -3xy² +9xy. The answer can be written as 9xy -3xy². [it is customery to write first the term which has + sign but there is nothing wrong even if term having - sign is written first]
SUM OF MORE THAN TWO SIMILAR OR DISSIMILAR TERMS
Let us suppose, there are five terms which are to be added. Then we write:
Sum of given terms = firest term + (second term) + (third term + (fourth term) + (fifth term)
Except first term, we put each term in small bracket and in between them we put + sign and then simplify the resultant expression by adding similar terms, if any.
Problem: Find sum of (i) -9xy² (ii) 3xy (iii) 4xy² (iv) x²y and (v) -xyz
Solution:
Sum of given terms = firest term + (second term) + (third term + (fourth term) + (fifth term)
On putting values of terms:
Sum of given terms= -9xy² + (3xy) + (4xy²) + (x²y) + (-xyz)
On removing brackets:
Sum of given terms= -9xy² + 3xy + 4xy² + x²y - xyz [except fifth term -xyz, all other tems within brackets have + sign. Therefore, multiplication of sign before the bracket with sign of term within the bracket has given + sign, whereas in case of fifth term, multiplication of + sign before the bracket with - sign of fifth term within the bracket has given - sign]
Now, variable parts of first and third term are same as xy². Therefore, on simplification, they will give single term. All other terms have different variable parts.
Hence sum of given terms=-9xy²+ 4xy² + 3xy + x²y - xyz
Or sum of given terms= -5 xy²+ x² + 3xy + xyz
Or sum of given terms= x² + -5 xy²+3xy + xyz [it is customary to put positive term first]
Resultant expression cannot be simplified further. Hence answer is x² + -5 xy²+3xy + xyz
ADDITION OF TWO EXPRESSIONS (WHETHER EACH OF THEM HAS MORE THAN ONE TERM OR ONE HAS MORE THAN ONE TERM AND OTHER HAS ONLY ONE TERM)
Procedure adopted for adding two expressions is the same as the procedure for adding two terms.
SUM OF TWO EXPRESSIONS = FIRST EXPRESSION + (SECOND EXPRESSION)
After putting both expressions in the above equation, we first remove bracket [for removing bracket, sign of each term within the bracket is multiplied by the sign appearing before bracket (here +)] and thereafter, we simplify the resultant expression by converting similar terms, if any, into a single term.
Problem. Find sum of (i) x² + 2xy + y² and (ii) 3x² - 2xy + y³
Solution:
Sum of two expressions = first expression + (second expression)
Here, sum of two given expressions =x² + 2xy + y² + (3x² - 2xy + y³)
On removing small bracket:
sum of two given expressions =x² + 2xy + y² + 3x² - 2xy + y³
Now we see that first and fourth terms have same variable part x² and second and fifth also have same variable part xy. Therefore on converting similar terms into single term, we get:
sum of two given expressions =4x² + y² + y³[ x² + 3x² have given 4x² and +2xy & - 2xy have given 0 xy or 0]
IN THE SIMILAR MANNER WE CAN ADD ANY NUMBER OF EXPRESSIONS TO GIVE A SINGLE EXPRESSION.

SUBTRACTION: Subtraction of one quantity from another quantity is inverse process of addition.
If we have to subtract second term from the first term
Difference of first term with second term = first term - (second term) In case of expressions: Difference of first espression with second expression = first expression - (second expression) After putting values of terms or expressions in the above, we solve the resultant expression in the same manner as in the case of addition of two terms or expressions. We should remember that first term is the term from which second term is to be subtradted. Similarly, first expression is the expression from which second expression is to be subtracted.
IMPORTANT: In case of subtraction, we always put second term (term which is to be subtracted) within a small bracket and a - sign before the small bracket.
Problem : Subtract -3xy from 9xy.
Solution: Here first term is 9xy because from 9xy, the other term -3xy is to be subtracted. Term -3xy is to be subtracted. Therefore, -3xy is the second term.
Difference of first term with second term = first term - (second term)
Hence difference of 9xy from -3xy = 9xy - (-3xy)
For removing the small bracket, we will have to multiply sign of term within the bracket with sign before the bracket. Here, sign of term within the bracket is - and sign before the bracket is also -. Therefore, their multiplication will give a+ sign.
On removing the bracket, we will get:
Difference = 9xy + 3xy
Now both terms are similar terms because their variable part xy is the same. Also both terms carry the same + sign, therefore,-
Difference = 12xy
Problem: Find difference of 3x² from 7x²
Solution: Here 3x² is to be subtracted from 7x². Therefore, 7x² is the first term and 3x² is the second term .
Now, Difference =first term - (second term) [do not forget to put second term within small bracket]
On putting values:
Difference = 7x² - (3x²)
On removing small bracket:
Difference = 7x² - 3x² [+ sign of 3x² on multiplication with - sign before the bracket has given - sign]
Variable parts of both terms are same as x<sup>2</sup>, therefore, they can be converted into single term. Therefore,
Difference= 4x²

Problem: Subtract x² + 3x +7 from 9x² - xy + 9
Here x² + 3x +7 is to be subtracted, therefore, it is second term. remaining term is first term.
Difference = first term - (second term)
On putting values, we get-
Difference=9x² - xy + 9 - (x² + 3x +7)
On removing the bracket, we get-
Difference=9x² - xy + 9 - x² - 3x -7 [Notice change of sign of each term within the bracket]
Now, first and fourth terms have same variable part, therefore they can be converted into single term. Also 9 and 7 are numbers and numbers can be added or subtracted independently if they are not coefficients to any variable, as may be required. Numbers used as coefficients can only be added or subtracted when their variable pats are the same.
Therefore, Difference= 8x² - xy -3x +2

THINGS TO REMEMBER:
(i)	While adding any number of terms, except first term all other terms are to be put in small brackets and a + sign is to be put in between them like Sum = first term + (second term) + (third term + (fourth term) + (fifth term)
(ii)	While subtracting second term from first term, second term is always to be put in small bracket and a - sign is to be put in beween them like Difference =first term - (second term). Second term is the term which is to be subtracted.
(iii)	While removing bracket, sign of term within the bracket gets multiplied by the sign immediately outside the bracket like -(3x) is -3x. Another example: (3x + 7y) is 3x +7y. In this case sign immediately outside bracket is +.
(iv)	Similar terms can be added or subtracted to give a single similar term
(v)	Terms which are not similar cannot be converted into single term.
(vi)	Two terms are said to be similar if their variable parts are same. For comparing variable parts of two terms, their complete part consisting of alphabets (variables) is to be taken together. They should have same number of alphabets (variables) and every variable should have same power. For example 3xy and 7xy are similar terms whereas 3x and 7xy are not similar terms. In this case, first term has variable part x & second term has variable part xy. Variable part of first term has one alphabet x as variable whereas variable part of second term has two alphabets x & y. Similarly, 3x²y & 3xy² are not similar terms. Their variable parts are x²y and xy². Though number of variable in both variable parts is 2 (first variable x & second variable y) yet their powers are different. Power of x in first term is 2 whereas in second term its power is 1. Similarly, in case of y, power of y in first term is 1 whereas in second term is 2. Therefore, these two terms x²y and xy² are not similar.
(vii)	While comparing power in variable parts of two terms, powers of each variable in first term are to be compared with powers of same variable separately in second term. For example, if variable parts of two terms 8x³yz & 9xy are to be comapred, then powers x in first term will be compared with power of x in second term, power of y in first term is to be compared with power of y in second term and power z in first term is to be compared with power of z in second term. Two terms will be similar only if (i) number of variable in both terms is same; and (ii) powers of each variable in both terms are same. Both conditions should be satisfied. If any condition is not satisfied, terms are not similar. In this case both conditions are not satisfied. Hence terms 8x³yz & 9xy are not similar.
(viii)	Numbers which are not associated with any variable (which are not coefficient of any term) may be added or subtracted (as may be required) to give a single number.
(ix)	If any term has variable part only (does not have any number as its coefficient) then its coefficient is 1.
(x)	Any term, as a whole, which does not have a - sign is positive in nature and in coputations its' sign is taken as +.
(xi)	Above all principles also apply in case of addition and subtraction of expressions.

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