System of Equations With Matrices: Background You’ll Need 1

  • Use the commutative, associative, and distributive properties of numbers to solve math problems.

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

[latex]a+b=b+a[/latex]

The commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

[latex]a\cdot b=b\cdot a[/latex]
Examples:

  • [latex]\left(-2\right)+7=5\text{ and }7+\left(-2\right)=5[/latex]
  • [latex]\left(-11\right)\cdot\left(-4\right)=44\text{ and }\left(-4\right)\cdot\left(-11\right)=44[/latex]

It is important to note that neither subtraction nor division is commutative.

Non-examples:

  • [latex]17 - 5[/latex] is not the same as [latex]5 - 17[/latex].
  • [latex]20\div 5\ne 5\div 20[/latex].

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

[latex]a\left(bc\right)=\left(ab\right)c[/latex]

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

[latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex]
Examples:

  • [latex]\left(3\cdot4\right)\cdot5=60\text{ and }3\cdot\left(4\cdot5\right)=60[/latex]
  • [latex][15+\left(-9\right)]+23=29\text{ and }15+[\left(-9\right)+23]=29[/latex]

Non-examples:

  • [latex]\begin{align}8-\left(3-15\right) & \stackrel{?}{=}\left(8-3\right)-15 \\ 8-\left(-12\right) & \stackrel{?}=5-15 \\ 20 & \neq 20-10 \\ \text{ }\end{align}[/latex]
  • [latex]\begin{align}64\div\left(8\div4\right)&\stackrel{?}{=}\left(64\div8\right)\div4 \\ 64\div2 & \stackrel{?}{=}8\div4 \\ 32 & \neq 2 \\ \text{ }\end{align}[/latex]

Note: neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

[latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex]

This property combines both addition and multiplication (and is the only property to do so).

Example:The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.Non-example:

[latex]\begin{align} 6+\left(3\cdot 5\right)& \stackrel{?}{=} \left(6+3\right)\cdot \left(6+5\right) \\ 6+\left(15\right)& \stackrel{?}{=} \left(9\right)\cdot \left(11\right) \\ 21& \ne 99 \end{align}[/latex]

Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.

A special case of the distributive property occurs when a sum of terms is subtracted.

[latex]a-b=a+\left(-b\right)[/latex]
Consider the difference [latex]12-\left(5+3\right)[/latex].
[latex]\\[/latex]
We can rewrite the difference of the two terms [latex]12[/latex] and [latex]\left(5+3\right)[/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\left(5+3\right)[/latex], we add the opposite.

[latex]12+\left(-1\right)\cdot \left(5+3\right)[/latex]

Now, distribute [latex]-1[/latex] and simplify the result.

[latex]\begin{align}12+\left(-1\right)\cdot\left(5+3\right)&=12+[\left(-1\right)\cdot5+\left(-1\right)\cdot3] \\&=12+(-5-3) \\&=12+\left(-8\right) \\&=4 \end{align}[/latex]

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

[latex]\begin{align}12-\left(5+3\right) &=12+\left(-5-3\right) \\ &=12+\left(-8\right) \\ &=4\end{align}[/latex]