# Properties for simplifying algebraic expressions

This lesson gives you a list of nine important properties for simplifying algebraic expressions.

Let a,b,and c represent real numbers.

Definition of subtraction :  a - b = a + (-b)

Definition of division : a ÷ b = a/b = a × 1/b, b ≠ 0

Multiplication by zero : 0 × a = 0

Multiplication by -1 : -1 × a = -a

Distributive property for subtraction : a(b - c) = ab - ac

Opposite of a difference : -(a - b) = b - a

Opposite of a sum : -(a + b) = -a + -b

Opposite of a product : -(ab) = -a × b =  a × -b

Opposite of an opposite : -(-a) = a

## Using the properties for simplifying algebraic expressions

Match the equation with the appropriate property name

A. 4(x - y) = 4x - 4y

B. (x - 2) ÷ 5z = (x - 2) / 5z

C. -(t-1) = -1(t-1)

D. 2d - 7 = 2d + (-7)

E. -( 2d - 7) = 7 - 2d

F. -[-(8 + s)] = 8 + s

G. -(12xy) = 12(-xy)

H. -[3 + (-a)] = -3 + [-(-a)]

I. (10x2 - 15y)(0) + -x = 0 + -x

1. Definition of division

2. Opposite of a sum

3. Opposite of an opposite

4. Opposite of a product

5. Multiplication by zero

6. Multiplication by -1

7. Opposite of a difference

8. Distributive property for subtraction

9. Definition of subtraction