22.1: Are They Equal?
Select all expressions that are equal to $8-12-(6+4)$.
- $8-6-12+4$
- $8-12-6-4$
- $8-12+(6+4)$
- $8-12-6+4$
- $8-4-12-6$
Let’s see how we can combine terms in an expression to write it with less terms.
Select all expressions that are equal to $8-12-(6+4)$.
Match each expression in column A with an equivalent expression from column B. Be prepared to explain your reasoning.
A
B
Write each expression with fewer terms. Show or explain your reasoning.
Combining like terms is an application of the distributive property. For example:
\(\begin{gather} 2x+9x\\ (2+9) \boldcdot x \\ 11x\\ \end{gather}\)
It often also involves the commutative and associative properties to change the order or grouping of addition. For example:
\(\begin{gather} 2a+3b+4a+5b \\ 2a+4a+3b+5b \\ (2a+4a)+(3b+5b) \\ 6a+8b\\ \end{gather}\)
We can't change order or grouping when subtracting; so in order to apply the commutative or associative properties to expressions with subtraction, we need to rewrite subtraction as addition. For example:
\(\begin{gather} 2a-3b-4a-5b \\ 2a+\text-3b+\text-4a+\text-5b\\ 2a + \text-4a + \text-3b + \text-5b\\ \text-2a+\text-8b\\ \text-2a-8b \\ \end{gather}\)
Since combining like terms uses properties of operations, it results in expressions that are equivalent.
The like terms that are combined do not have to be a single number or variable; they may be longer expressions as well. Terms can be combined in any sum where there is a common factor in all the terms. For example, each term in the expression $5(x+3)-0.5(x+3)+2(x+3)$ has a factor of $(x+3)$. We can rewrite the expression with fewer terms by using the distributive property:
\(\begin{gather} 5(x+3)-0.5(x+3)+2(x+3)\\ (5-0.5+2)(x+3)\\ 6.5(x+3)\\ \end{gather}\)