GENERATING EQUIVALENT EXPRESSIONS

An algebraic expression is a mathematical sentence involving constants (any real number), variables and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).

To generate equivalent expression to another expression, we have to be aware of the parts of an algebraic expression. 

We can use properties to combine like terms in an expression.

For example, let us consider the algebraic expression

3x + 2x + 4

You can  add / subtract the coefficients of the like terms to combine them. 

3x + 2x + 4  =  5x + 4

Write the equivalent expressions for the following :

Example 1 :

6x2 - 4x2

Solution :

6x2 - 4x2

= 2x2

Example 2 :

-3(5 - 6x)

Solution :

= -3(5 - 6x)

Use Distributive Property. 

= -3(5) - 3(-6x)

= -15 + 18x

= 18x - 15

Example 3 :

3a + 2(b + 5a)

Solution :

= 3a + 2(b + 5a)

Use Distributive Property. 

= 3a + 2b + 2(5a)

= 3a + 2b + 10a

= 13a + 2b

Example 4 :

 y + 11x + 7y - 7x

Solution :

= y + 11x + 7y - 7x

= 4x + 8y

Example 5 :

8m + 14 - 12 + 4n

Solution :

= 8m + 14 - 12 + 4n

= 8m + 4n + 2

Example 6 :

7(x - 3) + 2(2x - 5) - 3(x - 5)

Solution :

= 7(x - 3) + 2(2x - 5) - 3(x - 5)

Use Distributive Property.

= 7(x) + 7(-3) + 2(2x) + 2(-5) - 3(x) - 3(-5)

= 7x - 21 + 4x - 10 - 3x + 15

= 8x - 16

Example 7 :

4x - (2 + 4x) - 2(x - 1) - 8(x -3)

Solution :

4x - (2 + 4x) - 2(x - 1) - 8(x -3)

Use Distributive Property.

4x - 2 - 4x - 2(x) - 2(-1) - 8(x) - 8(-3)

4x - 2 - 4x - 2x + 2 - 8x + 24

= -10x + 24

Example 8 :

(x + 3)2 - (x - 3)2

Solution :

(x + 3)2 - (x - 3)2

(x + 3)(x + 3) - (x - 3)(x - 3)

(x2 + 3x + 3x + 9) - (x2 - 3x - 3x + 9)

(x2 + 6x + 9) - (x2 - 6x + 9)

x2 + 6x + 9 - x2 + 6x - 9

12x

Example 9 :

x2 + 5x - (x + 3)(x - 3)

Solution :

x2 + 5x - (x + 3)(x - 3)

x2 + 5x - (x2 - 3x + 3x - 9)

x2 + 5x - (x2 - 9)

x2 + 5x - x2 + 9

5x + 9

Example 10 :

25(x + 5) - 5(x + 5)2

Solution :

= 25(x + 5) - 5(x + 5)2

Factor (x + 5).

= (x + 5)[25 - 5(x + 5)]

= (x + 5)(25 - 5x + 25)

= (x + 5)(-5x)

= x(-5x) + 5(-5x)

= -5x2 - 25x

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