Combining Like Terms and Solving

When you get into long and complicated algebra equations, there are often multiple ways to solve them. 

One of the first things you should always do when working with equations, is combine like terms.  Combining like terms will not only make your equation shorter and easier to work with, but it will cut down on the number of steps you have to do when isolating a variable.

What are like terms? Like terms are terms that represent the same thing.  If you have the equation $5f+3f=19$, you can combine the terms with $f$.  Why?  Because one of the rules of algebra is that in any given equation, any variable only stands for one number.  So, we don't know what $f$ is, but we know that every $f$ is equal to the same number.  It's as if that sentence said "5 foxes plus 3 foxes."  We don't know much about the foxes, but we do know that we have 8 foxes.  Likewise, we can combine $5f+3f$ into $8f$.  One thing to pay attention to: $x$ and $x^2$ are not the same.  If $x=4$ then $x^2=16$ and $4$ and $16$ are definitely not the same number!

In equations, you combine all terms that are alike.

Numbers can be joined with other numbers (e.g., $3+2=5$). (Think about it, before you started dealing with variables, you combined like terms all the time!)

Terms with the same variable (raised to the same power) can be joined (e.g., $3x + 2x = 5x$ and $4x^2-x^2=3x^2$).

Before you try to solve an equation, combine all of the like terms on each side of the equal sign.   You can combine like terms even if they are separated by an unlike term.  You cannot combine terms that are on opposite sides of the equal sign!  When combining lots of terms, cross them out as you combine them, and rewrite your new combined equation after each pass, so that you do not lose any of your terms.

Example:

$$\require{cancel}\eqalign{3m + 2 - 4m + 6 &= 5m + 4 - 3m\quad&& \text{This equation has lots of like terms}\\\cancel{3m}+2 \cancel{-4m}+6&=5m + 4 -3m \quad&&\text{Combine the m terms on the left side}\\-m+6&=\cancel{5m}+4\cancel{-3m}\quad&&\text{Combine the m terms on the right side}\\-m+6&=2m+4\quad&&\text{All like terms combined}}$$

Now you just have to solve for the variable by isolating $m$:

$$\eqalign{-m+6&=2m+4\\+m\quad&=+m\quad&&\text{add }m \text{ to each side}\\6&=3m+4\\-4&=\qquad-4\quad&&\text{Subtract 4 from each side}\\2&=3m\\\div3&\quad\div3\quad&&\text{Divide both sides by 3}\\\dfrac{2}{3}&=m}$$

Overall, often ugly, many-termed problems like the one we started with can be simplified considerably by combining like terms.