College Preparatory Mathematics

Section 0.4 Properties of Algebra

In algebra we will often need to simplify an expression to make it easier to use. There are three basic forms of simplifying which we will review here.

World View Note: The term "Algebra" comes from the Arabic word al-jabr which means "reunion". It was first used in Iraq in \(830\) AD by Mohammad ibnMusa al-Khwarizmi.

The first form of simplifying expressions is used when we know what number each variable in the expression represents. If we know what they represent we can replace each variable with the equivalent number and simplify what remains using order of operations.

Whenever a variable is replaced with something, we will put the new number inside a set of parenthesis. Notice the \(3\) and \(5\) in the previous example are in parenthesis. This is to preserve operations that are sometimes lost in a simple replacement. Sometimes the parenthesis won’t make a difference, but it is a good habbit to always use them to prevent problems later.

It will be more common in our study of algebra that we do not know the value of the variables. In this case, we will have to simplify what we can and leave the variables in our final solution. One way we can simplify expressions is to combine like terms. Like terms are terms where the variables match exactly (exponents included). Examples of like terms would be \(3xy\) and \(7xy\) or \(3a^2b\) and \(8a^2b\) or \(3\) and \(5\text{.}\) If we have like terms we are allowed to add (or subtract) the numbers in front of the variables, then keep the variables the same. This is shown in the following examples

As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term, the sign always stays with the term.

A final method to simplify is known as distributing. Often as we work with problems there will be a set of parenthesis that make solving a problem difficult, if not impossible. To get rid of these unwanted parenthesis we have the distributive property. Using this property we multiply the number in front of the parenthesis by each term inside of the parenthesis.

Distributive Property: \(a(b + c) = ab + ac\)

Several examples of using the distributive property are given below.

In the previous example we again use the fact that the sign goes with the number, this means we treat the \(-6\) as a negative number, this gives \((-7)(-6) = 42\text{,}\) a positive number. The most common error in distributing is a sign error, be very careful with your signs!

It is possible to distribute just a negative through parenthesis. If we have a negative in front of parenthesis we can think of it like a \(-1\) in front and distribute the \(-1\) through. This is shown in the following example.

Distributing through parenthesis and combining like terms can be combined into one problem. Order of operations tells us to multiply (distribute) first then add or subtract last (combine like terms). Thus we do each problem in two steps, distribute then combine.

In the previous example we distributed \(-2\text{,}\) not just \(2\text{.}\) This is because we will always treat subtraction like a negative sign that goes with the number after it. This makes a big difference when we multiply by the \(5\) inside the parenthesis, we now have a positive answer. Following are more involved examples of distributing and combining like terms.