Properties or Laws of Addition and Multiplication

(updated 18 January 2022)

There are three major properties or laws concerning adding or multiplying expressions.

The commutative property says that the order in addition and multiplication does not matter. The associative property states that grouping location does not matter. The distributive property shows how multiplication of sums takes place.

Although these properties or laws may seem obvious, they are basics of Algebra operations.

This lesson will answer those questions.

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Commutative property

The commutative property states that expressions can be added or multiplied in any order.

x + y = y + x

xy = yx

This is obvious with numbers, since 23 + 7 = 7 + 23 and 5 × 8 = 8 × 5.

Larger number of expressions

The law can be extended to a larger number of expressions, as well as combinations of addition and multiplication:

uvw + x + y + z =

x + wvu + y + z =

z + x + vwu + y

Grouped expressions

It also applies to grouped expressions:

(x/2 + 4)(3y − 7) + z + 2 =

z + (3y − 7)(x/2 + 4) + 2

Subtraction and division

Although you can include subtraction and division in a group or parentheses, the order of subtraction and division is not commutative.

x − y ≠ y − x

Note: Think of x − y as x + (−y). Then, you can see that x − y = x + (−y) = −y + x.

Likewise,

x/y ≠ y/x

Associative property

The associative property states that when three or more expressions are added or multiplied, they may be grouped without affecting the answer. The commutative property applies within the associative property.

(x + y) + z = x + (y + z)

(xy)z = x(yz)

This law applies with a larger number of expressions, as well as grouped expressions.

(x + xy) + 3z + 5xz/2 =

x + (xy + 3z) + 5xz/2 =

x + (xy + 3z + 5xz/2)

Again, you should think of subtraction as addition of a negative number.

Distributive property

The distributive property states that multiplying an expression times the sum of expressions is the same as multiplying the expression times each item in the sum.

x(y + z) = xy + xz

A more complex example of the distributive property is:

(x − 3y)(z +5) =

z(x − 3y) + 5(x − 3y) =

xz − 3yz + 5x − 15y

Multiplying by a negative number

The distributive property works well when multiplying by a negative number.

−2x(y + 3) =

−2xy + (−2x)3 =

−2xy − 6x

Also,

−3x(y − 1) =

Change y − 1 to y + (− 1)

−3x[y + (−1)] =

−3xy + (−3x)(−1) =

−3xy + 3x

Factors

The reverse of the distributive law is factoring: x and (y + z) are factors of xy + xz.

(x − 3y) and (z +5) are factors of the xz − 3yz + 5x − 15y expression.

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Obey the law

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Resources and references

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