Simplify Algebraic Expressions

Simplify Algebraic Expressions

Simplify Algebraic Expressions

Simplify of algebraic expressions is the addition and subtraction of the terms of expressions. To simplify the algebraic expressions we need first to understand about the addition and subtraction of algebraic expressions.

Addition

In the addition of algebraic expression, we should add the coefficients of the like terms of the expression only. We can not add the coefficients of unlike terms.

For example:   
-     x + x + 2x = 4x
-     3y + 7y + 2z = 10y + 2z
-     3a2 + 5a2 + 2a = 8a2 + 2a

Subtraction

In the subtraction of algebraic expression, we should subtract the coefficients of the like terms of the expression only. We can not subtract the coefficient of unlike terms.

For example:
-     7a – 3a = 4a
-     5x2 – 3x2 = 2x2
-     4y2 – 2y – 3y2 = y2 – 2y

Addition and subtraction of two or more algebraic expressions

For the addition or subtraction of two or more than two algebraic expressions having two or more than two terms, we can add or subtract the terms of the algebraic expressions in vertical or horizontal arrangement. We should use the following steps:

-     Arrange the like terms in the same column at first.
-     After that we add or subtract their coefficients.

Study the following worked out examples:


Workout Examples

Example 1: Add: 3x2 + 5xy – 2y2 and 7x2 + 5y2 + 3xy

Solution: Addition by vertical arrangement:
Example 1: Addition by vertical arrangement

Addition by horizontal arrangement:
                        3x2 + 5xy – 2y2 + 7x2 + 5y2 + 3xy
                        = 3x2 + 7x2 + 5xy + 3xy – 2y2 + 5y2
                        = 10x2 + 8xy + 3y2


Example 2: Add: 4b2 + 3ab, 5a2 + 5ab and 3a2 + 6ab + 7b2

Solution: Addition by vertical arrangement:
Example 2: Addition by vertical arrangement

Addition by horizontal arrangement:
                        4b2 + 3ab + 5a2 + 5ab + 3a2 + 6ab + 7b2
                        = 4b2 + 7b2 + 3ab + 5ab + 6ab + 5a2 + 3a2
                        = 11b2 + 14ab + 8a2


Example 3: Subtract 3ab – 4bc + 8 from 7ab + 3bc - 4

Solution: Subtraction by vertical arrangement:
Example 3: Subtraction by vertical arrangement

Subtraction by horizontal arrangement:
                        7ab + 3bc – 4 – (3ab – 4bc + 8)
                        = 7ab + 3bc – 4 – 3ab + 4bc – 8
                        = 7ab – 3ab + 3bc + 4bc – 4 – 8
                        = 4ab + 7bc – 12


Example 4: What should be added to 3x + 4y + 5z to get 8x + y – 4z?

Solution: [ To solve this problem, let’s think first, what should be added to 6 to get 14? In 6, we should add 8 to get 14. And 8 is the difference of 14 and 6, i.e. 14 – 6 = 8. It is very good idea, and we use this idea in algebra also. So we should subtract 3x + 4y + 5z from 8x + y – 4z.]

            Now,
Example 4: Subtraction by vertical arrangement

The required expression to be added is 5x – 3y – 9z.


Example 5: What should be subtracted from 4x – 7y + 4z to get 2x + 5y + 5z?

Solution: [ To solve this problem, let’s think first, what should be subtracted from 9 to get 5? Obviously 4 should be subtracted. And 4 is the difference of 9 and 5, i.e. 9 – 5 = 4. It is very good idea, and we use this idea in algebra also. So we should subtract 2x + 5y + 5z from 4x – 7y + 4z.]

            Now,
Example 5: Subtraction by vertical arrangement

The required expression to be subtracted is 2x – 12y –z.


Example 6: Simplify: 10x2 – 2(x2 + 3x – 6) – x(x + 2)

Solution: Here,
                        10x2 – 2(x2 + 3x – 6) – x(x + 2)
                        = 10x2 – 2x2 – 6x + 12 – x2 – 2x
                        = 10x2 – 2x2 – x2 – 6x – 2x + 12
                        = 7x2 – 8x + 12


Example 7: Simplify: 2x(x – 3y) – 3y(x + 2y) – 2(x2 + xy)

Solution: Here,
                        2x(x – 3y) – 3y(x + 2y) – 2(x2 + xy)
                        = 2x2 – 6xy – 3xy – 6y2 – 2x2 – 2xy
                        = 2x2 – 2x2 – 6xy – 3xy – 2xy – 6y2
                        = – 11xy – 6y2


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