Mathematical Properties

Algebra I

Associative Property ofAddition and Multiplication

• The associative property means that you will get the same result no matter how you group the numbers that you add or multiply.

• Ex) (a + b) + c = a + (b + c) (3 + 4) + 8 = 3 + (4 + 8) (ab)c = a(bc) (3 · 4)8 = 3(4 · 8)• Change groups/ order stays the same.

Commutative Property of Addition and Multiplication

• The commutative property means that you can switch the order of the numbers that you add or multiply and still get the same result.

• Ex) x + y = y + x 7 + 4 = 4 + 7 xy = yx (7)(4) = (4)(7)

• Change the order.

Distributive Property

• The distributive property distributes the number you multiply by to the other numbers that are grouped in parentheses.

• Ex) a(b + c) = ab + ac (b + c)a = ba + ca 2(5 + 6) = 2(5) + 2(6) (5 + 6)2 = (5)(2) + (6)(2)

• Multiply when distributing.

Identity Property of Addition and Multiplication

• The identity property means that if you add 0 to a number or multiply a number by 1, the result is the same number.

• Ex) x + 0 = x and x · 1 = x 3 + 0 = 3 and 3 · 1 = 3

• The identity or value of the number stays the same.

Inverse Property of Addition

• The sum of an integer and its additive inverse is equal to zero.

• Ex) 3 + (-3) = 0 b + (-b) = 0

• Add the opposite.

Inverse Property of Multiplication

• Two numbers whose product is 1 are multiplicative inverses of each other.

• Ex) 4 . 5 = 1 and X . Y = 1 5 4 Y X

• Multiply by the reciprocal.

Multiplicative Property of Zero

• Any number multiplied by zero will take on the value of zero.

• Ex) 3 · 0 = 0 and x · 0 = 0

• Anything times zero is zero.

Reflexive Property

• Any quantity is equal to itself.

• Ex) a = a 7 = 7 2 + 3 = 2 + 3

Symmetric Property

• If one quantity equals a second quantity, then the second quantity equals the first.

• Ex) If a = b; then b = a If 9 = 6 + 3; then 6 + 3 = 9• If, then

Transitive Property

• If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity.

• Ex) If x = y and y = z; then x = z If 5 + 7 = 8 + 4 and 8 + 4= 12; then 5 + 7 = 12• If, and , then

Substitution Property

• A quantity may be substituted for its equal in any expression.

• Ex) If a = b, then a may be replaced for b in any expression.

If n = 15, then 3n = 3 · 15

• Used to solve many problems

Properties step by step

2 (3 · 2 – 5) + 3 · 1/3

Properties step by step

2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6

Properties step by step

2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6 2(1) + 3 · 1/3 substitution 6 – 5 = 1

Properties step by step

2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6 2(1) + 3 · 1/3 substitution 6 – 5 = 1 2 + 3 · 1/3 mult. identity 2 · 1 = 2

Properties step by step

2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6 2(1) + 3 · 1/3 substitution 6 – 5 = 1 2 + 3 · 1/3 mult. identity 2 · 1 = 2 2 + 1 mult. inverse 3 · 1/3 = 1

Properties step by step

2 (3 · 2 – 5) + 3 · 1/3 2(6 – 5) + 3 · 1/3 substitution 3 · 2 = 6 2(1) + 3 · 1/3 substitution 6 – 5 = 1 2 + 3 · 1/3 mult. identity 2 · 1 = 2 2 + 1 mult. inverse 3 · 1/3 = 1 3 substitution 2 + 1 = 3