For Grade 7:

Algerbra.

Constant / Base /Exponent:
In the expression,
3x5, -2y10
3 & -2 are constants,
x & y are bases,
5 & 10 are exponents or power.
If bases and exponents are same then we call them like terms. So 3x6 and 5x6 are like terms and they can be easily added and subtracted.

See more with examples (link to grade 6)

Simplification on expression:
If we have like terms in the expression then we can add/subtract them to simplify them.
For example:
3a + 9a + 4a + 5b
( 3a + 4a + 9a ) + 5b
so the simplified equation is:
16a + 5b.

4c + 7a + 5a + 5c
( 4c + 5c ) + ( 7a + 5a ),
so the expression after simplification becomes:
9c + 12a

Order of the algebraic expression:
The algebraic expression can have term including different powers (exponent). Depending on these exponents we arrange an algebraic expression in two ways.
· Ascending Order.
· Descending Order.

Ascending Order:
In this arrangement we arrange the term in an algebraic expression in increasing order of power.
For example:
The expression 3x2 + 5x6 +2y can be arranged as, 2y + 3x2 + 5x6.

Descending Order:
In this arrangement we arrange the terms of algebraic expression in decreasing order of power.
For example:
The expression 3x5 + 2y2 + 4y7 can be arranged as , 4y7 + 3x5 + 2y.

Operations on algebraic expressions:
· Addition of algebraic expression.
· Subtraction of algebraic expression.
· Multiplication of algebraic expression.
· Division of algebraic expression.

Addition:
Two expressions are added in such a way that like terms are added to the like terms. If there are no like terms then they are written in an order (mostly ascending order).
Examples:

1. Add 3a + 5c and 2a + 4c
3a + 5c
2a + 4c
5a + 9c

2. Add a + b + c and 2a + 3d + 4c
a + b + c
2a + 3d + 4c
3a + b + 5c + 3d
Here we can not add "b" to "3d" , as they are not like terms. Also notice as all the exponents are same that is "1" so we can write them in any sequence.

3. Add 2a + 3c and 3a2 + 4c3
2a + 3c
3a2 + 4c3
2a + 3c + 3a2 + 4c3
Here we can not add 2a with 3a2, they are not like term because their exponents are not same.

Subtraction:
In algebraic expression we define subtraction as another kind of addition. So x - y actually means x + (- y ). So we need to change the sign of the term to be subtracted.

Examples:

1. Subtract 5a + 7b + 2c and 3a + 4b + c
5a + 7b + 2c
+3a + 4b + c
_________
2a + 3b + c

2. Subtract 12a3 + 14b3 +15c3 and 5a3 +7b3 + 10c3.
12a3 + 14b3 + 15c3
+5a3 + 7b3 + 10c3
_________________
7a3 + 7b3 + 5c3

Multiplication:
For the multiplication we keep few points in mind.
(Note that we use the "." instead of multiplication sign "*" in algebra).
1. When the multiplication of any term is done with a constant , we multiply the constant of the term with the given constant.

Example:
3y2. 4 = 12y2.
2. When two terms of same variable are multiplied then their constants are multiplied and the exponents of the variables are added.

Example:
3y2 . 3y3 = 9y 2 + 3 = 9y5
3. When two terms of different variables are multiplied then it is in the following way,

Example:
3x2 . 4y3 = 12x2y3
The resultant term has the both variables in it.
4. When the whole expression is multiplied with another expression then the multiplication is done by multiplying each term of one expression with the whole other expression.

Example:
3a + 4b
* 2a + 5b
__________________
6a2 + 8ab
+ 15ab + 20b2
__________________
6a2 + 23ab + 20b2

Division:
For division also we keep few points in mind:

1. When an expression is divided with a constant, its power remains the same but its constant is effected.
Example:
15a7 / 3 = 5a7.

2. am / an = am - n that is when the bases are same and the term are divided then the powers are subtracted.
Example:
4x2 / 2x = 2 x 2 - 1 = 2x

3. When an expressions is divided with a term then each term of the expression is divided with the term.
Example:
24a3b4 + 30a5b5 + 12a6b4 / 6a2b2
= 24a3b4 / 6a2b2 + 30a5b5 / 6a2b2 + 12a6b4 / 6a2b2
= 4a3-2b4-2 + 5a5-3b5-2 + 2a6-2b4-2
= 4ab2 + 5a2b3 + 2a4b2

 

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