1. Fundamentals of Algebra
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2. 2.2 Simplifying Algebraic Expressions

3. What You Will Learn Use the properties of algebra.
Combine like terms of an algebraic expression.
Simplify an algebraic expression by rewriting the terms.
Use the Distributive Property to remove symbols of grouping.

4. Properties of Algebra

5. Review of Properties of Real Numbers
Properties of Algebra
You are now ready to combine algebraic expressions using the properties of real numbers.
Review of Properties of Real Numbers
1. Commutative Property of Addition:
Two real numbers can be added in either order.
a + b = b + a = 5 + 3
2. Commutative Property of Multiplication:
Two real numbers can be multiplied in either order.
ab = ba 4 •(-7) = -7 • 4

6. Properties of Algebra 3. Associative Property of Addition:
When three real numbers are added, it makes no different which two are added first.
(a + b) + c = a + (b + c) (2 + 6) + 5 = 2 + (6 + 5)
4. Associative Property of Multiplication:
When three real numbers are multiplied, it makes no difference which two are multiplied first.
(ab)c = a(bc) (3 • 5) • 2 = 3 • (5 • 2)
5. Distributive Property:
Multiplication distributes over addition.
a(b + c) = ab + ac 3(8 + 5) = 3 • • 5
(a + b)c = ac + bc (3 + 8)5 = 3 • • 5

7. Properties of Algebra 6. Additive Identity Property:
The sum of zero and a real number equals the number itself.
a + 0 = 0 + a = a = = 3
7. Multiplicative Identity Property:
The product of 1 and a real number equals the number itself
a • 1 = 1 • a = a 4 • 1 = 1 • 4 = 4
8. Additive Inverse Property:
The sum of a real number and its opposite is zero
a + (-a) = (-3) = 0
9. Multiplicative Inverse Property:
The product of a nonzero real number and it’s reciprocal is 1

8. Example 1 – Applying Properties of Real Numbers
Use the indicated rule to complete each statement.
a. Additive Identity Property: (x – 2) = x – 2
b. Commutative Property of Multiplication: 5(y + 6) =
c. Commutative Property of Addition: (y + 6) =
d. Distributive Property: (y + 6) =
e. Associative Property of Addition: (x2 + 3) + 7 =
f. Additive Inverse Property: x = 0

9. Example 1 – Applying Properties of Real Numbers
cont’d
a. (x – 2) + 0 = x – 2
b. 5(y + 6) = (y + 6)5
c. 5(y + 6) = 5(6 + y)
d. 5(y + 6) = 5y + 5(6)
e. (x2 + 3) + 7 = x2 + (3 + 7)
f. –4x + 4x = 0

10. Example 2 – Identifying the Terms of Expressions
Use the Distributive Property to expand each expression.
a. 2(7 – x) b. (10 – 2y)3 c. 2x(x + 4y)
d. –(1 – 2y + x)
Solution:
a. 2(7 – x) = 2  7 – 2  x
= 14 – 2x
b. (10 – 2y)3 = 10(3) – 2y(3)
= 30 – 6y

11. Example 2 – Identifying the Coefficients of Terms
cont’d
c. 2x(x + 4y) = 2x(x) + 2x(4y)
= 2x2 + 8xy
d. –(1 – 2y + x) = (–1)(1 – 2y + x)
= (–1)(1) – (–1)(2y) + (–1)(x)
= –1 + 2y – x

12. Example 3 – Geometry: Visualizing the Distributive Property
Write the area of each component of each figure. Then demonstrate the Distributive Property by writing the total area of each figure in two ways.

13. Example 3 – Geometry: Visualizing the Distributive Property
cont’d
The total area is 3(2 + 4) = 3 • • 4 =
= 18 6 12
The total area is a(a + b) = a • a + a • b
= a2 + ab a2 ab

14. Example 4 – Using Mental Math in Everyday Applications
You can earn $14 per hour and time-and-a-half for overtime. Show how you can use the Distributive property to find your overtime wage mentally.
Solution:
You can think of the “time-and-a-half” as So, using the Distributive Property, you can think of the following without writing anything on paper.

15. Example 4 – Using Mental Math in Everyday Applications
cont’d
b. You are buying 15 potted plants that cost $19 each. Show how you can use the Distributive Property to find the total cost mentally.
Solution:
b. You can think of the $19 as $20 – $1. So, using the Distributive Property, you can think of the following without writing anything on paper.
15(19) = 15(20 – 1)
= 15(20) – 15(1)
= 300 – 15
= $285

16. Combining Like Terms

17. Combining Like Terms
In an algebraic expression, two terms are said to be like terms if they are both constant terms or if they have the same variable factor(s).
Factors such as x and 5x and ab in 6ab are called variable factors.
The terms 5x and –3x are like terms because they have the same variable factor, x.
Similarly, 3x2y, –x2y, and (x2y) are like terms because they have the same variable factors, x2 and y.

18. Example 5 – Identifying Like Terms
Expression Like Terms
a. 5xy + 1 – xy 5xy and –xy
b. 12 – x2 + 3x – and –5
c. 7x – 3 – 2x x and –2x, –3 and 5

19. Combining Like Terms
To combine like terms in an algebraic expression, you can simply add their respective coefficients and attach the common variable factor(s).
This is actually an application of the Distributive Property, as shown in Example 5.

20. Example 6 – Combining Like Terms
Simplify each expression by combining like terms.
a. 5x + 2x – 4 b. – y – 5y
c. 2y – 3x – 4x d. 7x + 3y – 4x
e. 12a – 5 – 3a + 7
Solution
a. 5x + 2x – 4 = (5 + 2)x – 4
= 7x – 4
b. – y – 5y = (–5 + 8) + (7 – 5)y
= 3 + 2y
Distributive Property
Add.
Distributive Property
Simplify.

21. Example 6 – Combining Like Terms
cont’d
c. 2y – 3x – 4x = 2y – x(3 + 4)
= 2y – x(7)
= 2y – 7x
d. 7x + 3y – 4x = 3y + 7x – 4x
= 3y + (7x – 4x)
= 3y + (7 – 4)x
= 3y + 3x
e. 12a – 5 – 3a + 7 = 12a – 5 – 3a + 7
= (12a – 3a) + (–5 +7)
= (12 – 3)a + (–5 +7)
= 9a + 12
Distributive Property
Add.
Simplify.
Commutative Property
Associative Property
Distributive Property
Subtract
Commutative Property
Associative Property
Distributive Property
Simplify

22. Simplifying Algebraic Expressions

23. Simplifying Algebraic Expressions
Simplifying an algebraic expression by rewriting it in a more usable form is one of the most frequently used skills in algebra.
To simplify an algebraic expression generally means to remove symbols of grouping and combine like terms.
For instance, the expression x + (3 + x) can be simplified as 2x + 3.

24. Example 7 – Simplifying Algebraic Expressions
Simplify each expression.
a. –3(–5x) b. 7(–x) c.
d. x2(–2x3) e. (–2x)(4x) f. (2rs)(r2s)
Solution
a. –3(–5x) = (–3)(–5)x
= 15x
b. 7(–x) = 7(–1)(x)
= –7x
Associative Property
Multiply.
Coefficient of –x is –1.
Multiply.

25. Example 7 – Simplifying Algebraic Expressions
cont’d
Solution c.
= 1 • x
= x
d. x2(–2x3) = (–2)(x2 • x3)
= –2 • x • x • x •x • x
= –2x5
Coefficient of
Commutative and Associative
Properties
Multiplicative Inverse
Multiplicative Identity
Commutative and Associative
Properties
Repeated Multiplication
Exponential Form

26. Example 7 – Simplifying Algebraic Expressions
cont’d
Commutative and Associative
Properties
e. (–2x)(4x) = (–2 • 4)(x •x)
= –8x2
f. (2rs)(r2s) = 2(r • r2)(s • s)
= 2 • r • r • r • s • s
= 2r3s2
Exponential Form
Commutative and Associative
Properties
Repeated multiplication
Exponential Form

27. Symbols of Grouping

28. Example 8 – Removing Symbols of Grouping
The main tool for removing symbols of grouping is the Distributive Property, as illustrated in this example.
Simplify each expression.
a. –(3y + 5) b. 5x + (x – 7)2
c. –2(4x – 1) + 3x d. 3(y – 5) – (2y – 7)

29. Example 8 – Removing Symbols of Grouping
cont’d
a. –(3y + 5) = –3y – 5
b. 5x + (x – 7)2 = 5x + 2x – 14
= 7x – 14
c. –2(4x – 1) + 3x = –8x x
= –8x + 3x + 2
= –5x + 2
d. 3(y – 5) – (2y – 7) = 3y – 15 – 2y + 7
= (3y – 2y) + (–15 + 7)
= y – 8
Distributive Property
Distributive Property
Combine like terms.
Distributive Property
Commutative Property
Combine like terms.
Distributive Property
Group like terms.
Combine like terms.

30. Example 9 – Geometry: Writing and Simplifying a Formula
Write and simplify an expression for (a) the perimeter and (b) the area of the triangle.

31. Example 9 – Geometry: Writing and Simplifying a Formula
cont’d
Solution:
a. Perimeter of a Triangle = Sum of the Three Sides
= 2x + (2x + 4) + (x + 5)
= (2x + 2x + x) + (4 + 5)
= 5x + 9
b. Area of a Triangle = • Base • Height
Substitute
Group like terms
Combine like terms
Substitute
Commutative Property
Multiply
Distributive Property

32. Example 10 – Geometry: Writing and Simplifying a Formula
The formula for the area of a trapezoid is
Use this formula to write and simplify an expression for the area of the proposed trapezoidal park.

33. Example 10 – Geometry: Writing and Simplifying a Formula
cont’d
Solution:
Begin by assigning the follow values.
h = x mi b1 = (x + 0.5) mi b2 = (3x + .03) mi
Then use the formula to write an expression.