SIMPLIFY VARIABLE EXPRESSIONS USING PROPERTIES

Simplify the following variable expressions :

Example 1 :

(y + 3) + 8

Solution :

= (y + 3) + 8

= y + 11

Example 2 :

3(4m)

Solution :

= 3(4m)

= 12m

Example 3 :

(2 + x) + 5

Solution :

= (2 + x) + 5

= x + 5

Example 4 :

(7p)(5)

Solution :

= (7p)(5)

= 35p

Example 5 :

4(2x + 3 - 5x) + 2(3x - 4)

Solution :

= 4(2x + 3 - 5x) + 2(3x - 4)

Apply distributive property.

= 8x + 12 - 20x + 6x - 8

Combine the like terms.

= 8x + 6x - 20x + 12 - 8

= -6x + 4

Example 6 :

5(2x - 3) + 7(2 - 5x) - (3x - 4)

Solution :

= 5(2x - 3) + 7(2 - 5x) - (3x - 4)

Apply distributive property.

= 10x - 15 + 14 - 35x - 3x + 4

Combine the like terms.

 = 10x - 35x - 3x - 15 + 14 + 4

= -28x + 3

Example 7 :

-8(-5b + 7) + 5b - 3b

Solution :

= -8(-5b + 7) + 5b - 3b

Apply distributive property.

= 40b - 56 + 5b - 3b

Combine the like terms.

= 40b + 5b - 3b - 56

= 42b - 56

Example 8 :

-4p - (1 - 6 p)

Solution :

= -4p - (1 - 6p)

Apply distributive property.

= -4p - 1 + 6p

Combine the like terms.

= -4p + 6p - 1

= 2p - 1

Example 9 :

4 - 5(-4n + 3) - (3 - 2n)

Solution :

= 4 - 5(-4n + 3) - (3 - 2n)

Apply distributive property.

= 4 + 20n - 15 - 3 + 2n

Combine the like terms.

= 20n  + 2n - 15 - 3 + 4

= 22n - 14

Example 10 :

 2(x + y) + 3(2x + 4y)

Solution :

= 2(x + y) + 3(2x + 4y)

Apply distributive property.

= 2x + 2y + 6x + 12y

Combine the like terms.

= 2x + 6x + 2y + 12y

= 8x + 14y

Example 11 :

Write an expression in simplest form that represents the perimeter of the polygon.

Solution :

Perimeter of rectangle = 2(length + width)

lenght = 8 cm and width = x cm

Perimeter of rectangle = 2(8 + x)

Using distributive property, we get

= 16 + 2x

So, the perimeter of the rectangle is 16 + 2x.

Example 12 :

Each runner is carrying an 8-ounce bottle of water, a 2.1-ounce energy bar, and a 3-ounce energy drink.

Write an expression in simplest form that represents the weight carried by y runners. Interpret the expression.

Solution :

Weight of water botle = 8 ounce

Weight of energy bar = 2.1 ounce

Weight of energy drink = 3 ounces

Total weight carried by him = 8 + 2.1 + 3

= 13.1 ounce

Example 13 :

John weighs 65 kilograms, Sam weighs 22x kilograms, and Mark weighs 13x kilograms. Write an expression in simplest form for their combined weight.

Solution :

Weight of John = 65 kg

Weight of Sam = 22x kg

Weight of Mark = 13x kg

Combined weight = 65 + 22x + 13x

= (65 + 35x)

Example 14 :

Are the expressions 8x² + 3 (x² + y) and 7x² + 7y + 4x² − 4y equivalent? Explain your reasoning

Solution :

Voth are equal.

Example 15 :

Are the expressions 8a² − 4b + 7a² and 15a² − 10b + 6b equivalent? Explain your reasoning.

Solution :

So, both are equal.

Example 16 :

The menu below shows the prices at Lunchtime Café. Lucita orders a turkey sandwich and two fruit cups. What expression should she use to determine the cost of her meal?

a)  4.50 + (2 x 2.50)       b)  2.50 + (2 x 4.50)

c)  4.50 + 2.50        d)  2.50 x 4.50

Solution :

Cost of Turkey Sandwich = $4.50

Cost of fruit cup = $2.50

Number of fruit cups he is purchasing = 2

Total cost = 4.50 + 2 x 2.50

So, option a is correct.

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