SIMPLIFYING ALGEBRAIC EXPRESSIONS

Problem 1 :

(4x + 7x)3

Solution :

= (4x + 7x)3

Distributing 3,

= [4x(3) + 7x(3)]

= 12x + 21x

= 33x

Problem 2 :

12z – 5z + 9z²

Solution :

= 12z – 5z + 9z²

= 7z + 9z²

Factoring z,

= 9z² + 7z

= z(9z + 7)

Problem 3 :

-7(-6m + 11m)

Solution :

= -7(-6m + 11m)

= -7(5m)

= -35m

Problem 4 :

4(11 – 3x)

Solution :

= 4(11 – 3x)

Distributing 4,

= 4(11) + 4(-3x)

= 44 – 12x

= -12x + 44

Problem 5 :

-5(5a – 3b - 6)

Solution :

= -5(5a – 3b - 6)

Distributing -5, 

= -5(5a) -5(-3b) -5(-6)

= -25a + 15b +30

Problem 6 :

-2(x² - 8x + 3x³ - 6)

Solution :

= -2(x² - 8x + 3x³ - 6)

= -2(x²) - 2(-8x) - 2(3x³) - 2(-6)

= -2x² +16x – 6x³ + 12

= -6x³ - 2x² + 16x + 12

Problem 7 :

9x – 4(6 – 3x)

Solution :

= 9x – 4(6) -4(-3x)

= 9x – 24 + 12x

combining like terms,

= 21x - 24

Problem 8 :

 5(3b – 2a) – 7b

Solution :

= 5(3b) + 5(-2a) – 7b

= 15b – 10a -7b

combining like terms,

= -10a +15b – 7b

= -10a + 8b

Problem 9 :

12 + 3(7x + 2)

Solution :

= 12 + 3(7x) + 3(2)

= 12 + 21x + 6

= 21x + 18

Problem 10 :

6(4y + 3z) – 11z

Solution :

= 6(4y) + 6(3z) – 11z

= 24y + 18z – 11z

= 24y – 7z

Problem 11 :

5 + 2(4m – 7n) + 9n

Solution :

= 5 + 2(4m) + 2(-7n) + 9n

= 5 + 8m – 14n + 9n

= 5 + 8m - 5n

Problem 12 :

12 – 7(3 – 5r) + 8r

Solution :

= 12 – 7(3) -7(-5r) + 8r

= 12 – 21 + 35r + 8r

= -9 + 43r

Problem 13 :

A class for 30 students visit an art museum and a special exhibit wile here.

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

b) Estimate a reasonable value of x. Explain.

c)  Use your estimate for x to evaluate the original expression and the simplified expression in part

(a). Are the values the same?

Solution :

a) Amount spent by each student to visit Museum and Exhibit = 8 + x

Amount spent by all 30 students = 30 (8 + x)

Using distributive property, we get

= 30(8) + 30(x)

= 240 + 30x

b) x is the cost of each ticket to visit exhibit, it must be lesser than cost of ticket to visit Museum. The reasonable value of x is $5.

c)  Applying x = 5, we get

= 240 + 30(5)

= 240 + 150

= $390

Applying x = 5 in 30(8 + x)

= 30(8 + 5)

= 30(13)

= $390

Problem 14 :

The original price of a model car is d dollars. You use a coupon and buy the kit for (d − 10) dollars. You assemble the model car and sell it for (2d − 20) dollars. Write an expression that represents your earnings. Interpret the expression

Solution :

Amount you earn = selling price - amount for coupon

= 2d - 20 - (d - 10)

= 2d - 20 - d + 10

= 2d - d - 20 + 10

= d - 10

(d − 10); You earn (d − 10) dollars. You also paid (d − 10) dollars, so you doubled your money by selling the model car for twice as much as you paid for the kit.

Problem 15 :

Each day, you run on a treadmill for r minutes and lift weights for 15 minutes. Which expressions can you use to find how many minutes of exercise you do in 5 days? Explain your reasoning.

a)  5(r + 15)      b)  5r + 5 ⋅ 15     c)  5r + 15     d)  r (5 + 15)

Solution :

Time spent on treadmill = r minutes

Time spent on lift weights = 15 minutes

Total time spent on both = r + 15

Number of days = 6

Number of minutes to exercise in these 5 days = 5(r + 15)

So, option a is correct.

Problem 15 :

A cheetah can run 103 feet per second. A zebra can run x feet per second. Use the Distributive Property to write and simplify an expression for how much farther the cheetah can run in 10 seconds.

Solution :

Speed of cheetah = 103 feet per second

Speed of zebra = x feet per second

Speed of cheetah is more than speed fo zebra.

Difference between them = 103 - x

In 10 second = 10(103 - x)

= 10(103) - 10 (x)

= 1030 - 10x