EQUIVALENT EXPRESSIONS

Equivalent expressions are algebraic expressions that will work the same even if they look different. Two algebraic expressions are considered to be equivalent, if they have the same value, when we substitute in the same value(s) for the variable(s).

Examples 1 :

In (-5 + 3) and (3 - 5), the numbers  are presented in different orders, but both are equal to -2.

-5 + 3 = -2

3 - 5 = -2

So, (-5 + 3) and (3 - 5) are equivalent operations.

Examples 2 :

In 1/2 and 3/6, the numbers  are different, both are equal to 0.5.

1/2 = 0.5

3/6 = 0.5

So, 1/2 and 3/6 are equivalent fractions.

Examples 3 :

Consider the two expressions given below.

3(x + 2)

3x + 6

If you substitute the any and same real value for x in both the expressions, both of them will result the same value.

When x = -1,

3(-1 + 2) = 3(1) = 3

3(-1) + 6 = -3 + 6 = 3

When x = 0,

3(0 + 2) = 3(2) = 6

3(0) + 6 = 0 + 6 = 6

When x = 2,

3(2 + 2) = 3(4) = 12

3(2) + 6 = 6 + 6 = 12

3(x + 2) = 3x + 6 for all real values of x

Therefore 3(x + 2) and 3x + 6 are equivalent algebraic expressions.

Generating Equivalent Algebraic Expressions

Write the equivalent algebraic expressions for the following :

Example 1 :

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 2 :

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 3 :

Solution :

Example 4 :

Solution :

Example 5 :

Solution :

Example 6 :

Solution :

Example 7 :

Solution :

Example 8 :

Solution :

Example 9 :

Write and simplify expressions for the area and perimeter of the rectangle.

Solution :

Length = 12

Width = 5.5 + x

Area of rectangle = length (width)

= 12(5.5 + x)

Using distributive property,

= 66 + 12x

Perimeter of rectangle = 2(length + width)

= 2(5.5 + x + 12)

= 2(17.5 + x)

Using distributive property, we get

= 34 + 2x

Example 10 :

Solution :

Length = 7 + x + 5

= 12 + x

Width = 9

Area of rectangle = length (width)

= (12 + x)9

= 9(12 + x)

Using distributive property,

= 108 + 9x

Perimeter of rectangle = 2(length + width)

= 2(12 + x + 9)

= 2(21 + x)

Using distributive property, we get

= 42 + 2x

Example 11 :

An art club sells 42 large candles and 56 small candles.

a)  Use the distributive property to write and simplify an expression for the profit.

b)  A large candles cost $5 and a small candle costs $3. What is the club's profit ?

Solution :

• Number of large candles = 42
• Cost of each large candle = x
• Number of small candles = 56
• Cost of each small candle = y

Profit for each large candle = 10 - x

Profit for each small candle = 5 - y

a)  Total profit of selling large and small candle

= 42(10 - x) + 56(5 - y)

= 420 - 42x + 280 - 56y

= 700 - 42x - 56y

b) When x = 5 and y = 3

Club's profit = 700 - 42(5) - 56(3)

= 700 - 210 - 168

= 700 - 378

= 322

Example 12 :

Write and simplify an expression for the difference between the perimeters of the rectangle and the hexagon. Interpret your answer.

Solution :

Length of rectangle = 2x + 7

Width of rectangle = 2x

Perimeter of rectangle = 2(length + width)

= 2(2x + 7 + 2x)

= 2(4x + 7)

= 8x + 14

Perimeter of hexagon = x + x + 8 + x + 6 + x + 2x + 2x

= 7x + 14

Difference between the perimeters = 8x + 14 - (7x + 14)

= 8x - 7x + 14 - 14

= x

Example 13 :

Add one set of parentheses to the expression

7 ⋅ x + 3 + 8 ⋅ x + 3 ⋅ x + 8 − 9

so that it is equivalent to 2(9x + 10)

Solution :

= 7 ⋅ (x + 3) + 8 ⋅ x + 3 ⋅ x + 8 − 9

= 7x + 21 + 8x + 3x + 8 - 9

= 18x + 29 - 9

= 18x + 20

Example 14 :

Which expression is not equivalent to 16x + 24?

a)  2(8x + 12)    b)  4(4x + 6)     c)  6(3x + 4)  d)  (2x + 3)8

Solution :

So, option c is not correct.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.