# Combining Like Terms

Like or similar terms can be roughly defined as algebraic expressions that share the same variables and identical exponents. The coefficients do not need to match. If the terms are like, we can combine them through addition or subtraction.

The most effective method to understand like terms is by studying examples.

These are some examples of like/similar terms.

\( \Rightarrow \) \(3\) and \(-5\) are like terms because they are both constants.

\( \Rightarrow \) \(7x\) and \(2x\) are like terms because they have the same variable \(x\).

\( \Rightarrow \) \( – 4{y^2}\) and \(9{y^2}\) are like terms because not only they have the same variable \(y\) but the values of their exponents are also the same which is \(2\).

\( \Rightarrow \) \(6a{b^2}\) and \( – 2a{b^2}\) are like terms because they have common group of variables and the exponent of each variable matches.

**Example 1:** Combine the like terms below.

$$a + 2b + 3c$$

The variables \(a\), \(b\), and \(c\) are different variables therefore they are NOT like terms. That means we can’t combine them.

**Example 2:** Combine the like terms below.

$$3x – 5y + x + 4y$$

Notice that \(3x\) and \(x\) are like terms. In the same way, \(-5y\) and \(4y\) are like terms. Since they are like terms, we can combine them by adding or subtracting. We can put them side by side for clarity. Thus, we have

$$3x + x – 5y + 4y$$

$$4x – y$$

**Example 3:** Combine the like terms below.

$$6mn\, – \,m{n^2} \,- \,{m^2}n \,- \,5mn + 3m{n^2} – {m^2}n$$

What we can do is to put the like terms side by side. Then, we operate them by adding or subtracting.

$$6mn \,- \,5mn = mn$$

$$ – m{n^2} + 3m{n^2} = 2m{n^2}$$

$$ – {m^2}n\, – \,{m^2}n = – 2{m^2}n$$

We can put them together to write our final answer.

$$mn + 2m{n^2} \,- \,2{m^2}n$$

**Example 4:** Simplify by combining like terms.

$$10 \,- \,{r^2}\, – \,4j{r^2}\, -\, 7 + 3{r^2} + 4j{r^2} + jr$$

Just like in our previous example, let’s put the like terms side by side.

The numbers are like terms.

$$10 \,-\, 7 = 3$$

The \(r^2\)’s are like terms.

$$ – {r^2} + 3{r^2} = 2{r^2}$$

The \(jr^2\)’s are like terms.

$$ – 4j{r^2} + 4j{r^2} = 0$$

The \(jr\) is by itself. Nothing to combine with.

$$jr$$

Putting them back together, we have

$$2{r^2} + jr + 3$$