Combining like terms allows us to write expressions more simply with fewer terms. But it can sometimes be tricky with long expressions, parentheses, and negatives. It is helpful to think about some common errors that we can be aware of and try to avoid:
- $6x-x$ is not equivalent to 6. While it might be tempting to think that subtracting $x$ makes the $x$ disappear, the expression is really saying take 1 $x$ away from 6 $x$'s, and the distributive property tells us that $6x-x$ is equivalent to $(6-1)x$.
- $7-2x$ is not equivalent to $5x$. The expression $7-2x$ tells us to double an unknown amount and subtract it from 7. This is not always the same as taking 5 copies of the unknown.
- $7-4(x+2)$ is not equivalent to $3(x+2)$. The expression tells us to subtract 4 copies of an amount from 7, not to take $(7-4)$ copies of the amount.
If we think about the meaning and properties of operations when we take steps to rewrite expressions, we can be sure we are getting equivalent expressions and are not changing their value in the process.