Previously in this unit, students solved equations of the form \(px+q=r\) and \(p(x+q)=r.\) Sometimes, work has to be done on a more complicated expression to get an equation into one of these forms. And sometimes, it is desirable to rewrite an expression in an equivalent form to understand how the quantities it represents are related. This work has some pitfalls when the expression has negative numbers or subtraction. For example, it is common for people to rewrite \(6x-5+2x\) as \(4x+5\) by reading “\(6x\) minus” and so subtracting the \(2x\) from the \(6x.\) Another example is rewriting an expression like \(5x-2(x+3)\) as \(5x-2x+6.\) Students do not see expressions as complicated as these in this lesson (they are coming in the next few lessons), but this lesson is meant to inoculate students against errors like these by reminding them that while subtraction is not commutative, addition is, and subtraction can be rewritten as adding the opposite. So in our example, \(6x-5+2x\) can be rewritten \(6x + \text-5 + 2x\) and then rearranged \(6x+2x+\text-5.\) Likewise, \(5x-2(x+3)\) can be rewritten \(5x + \text-2(x+3)\) before distributing -2.
- Explain (orally, in writing, and using other representations) how the distributive and commutative properties apply to expressions with negative coefficients.
- Justify (orally and in writing) whether expressions are equivalent, including rewriting subtraction as adding the opposite.
Let's find ways to work with subtraction in expressions.
- I can organize my work when I use the distributive property.
- I can re-write subtraction as adding the opposite and then rearrange terms in an expression.
A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression \(5x + 18\) has two terms. The first term is \(5x\) and the second term is 18.