Combining Like Terms

🧩 Combining Like Terms

Part 1 of 5 — What Is a Term, and What Makes Terms "Like"?

Topics in This Part

🔑 Key Concept: Combining like terms is just counting. If $3$ apples plus $2$ apples is $5$ apples, then $3x + 2x = 5x$. The whole skill rests on knowing which terms are apples — that's what Part 1 teaches.

Terms, Coefficients, and Variables

An expression is a string of terms joined by $+$ and $-$ signs. A term is a single number, a single variable, or a number times variables.

$\underbrace{7x}_{\text{term}} \;+\; \underbrace{4}_{\text{term}} \;-\; \underbrace{2y}_{\text{term}}$

Every term has two parts:

• the coefficient — the number multiplied in front, and
• the variable part — the letters (and their exponents).

🔑 Two quiet rules: A lone variable like $x$ has coefficient $1$ (we just don't write it). The sign in front belongs to the term — in $8a - 3b$, the second term is $-3b$.

Concept Check 🎯

The Rule: Same Variable, Same Exponent

Two terms are like terms when their variable parts are identical — exactly the same letters raised to exactly the same powers. The coefficients do not have to match.

$\boxed{\;\text{Like terms} \Longleftrightarrow \text{identical variable part}\;}$

⚠️ Watch the exponent. $x^2$ and $x$ are not like terms, even though they share the letter $x$. The power must match too — $x^2$ counts "squares," $x$ counts "singles."

Like or Not Like? 🔽

For each pair, decide whether the two terms are like terms.

Concept Check 🎯

What You Can Now Do

You can pick an expression apart into terms, read off each coefficient (remembering the hidden $1$ and the sign), and sort terms into like-groups.

💡 Think of like terms as labeled boxes: one box for $x$'s, one for $y$'s, one for plain numbers. You can only add what goes in the same box. In Part 2 we actually add them up.

🧩 Combining Like Terms

Part 2 of 5 — Adding the Coefficients

🔑 The Idea: To combine like terms, add (or subtract) the coefficients and keep the variable part exactly the same. The variable just rides along — it never changes.

The One Move

$3x + 5x = (3 + 5)\,x = 8x$

You added the coefficients $3$ and $5$ to get $8$, and the $x$ stayed put. This is the distributive property running in reverse — but you can just think of it as counting:

$3$ x's plus $5$ x's is $8$ x's.

More Examples

⚠️ Never multiply the variables. $3x + 5x$ is $8x$, not $8x^2$. You are counting x's, not multiplying them. ($x^2$ would only appear from $3x \cdot 5x$.)

Concept Check 🎯

Keeping Track of Signs

The trick is to treat each term with the sign in front of it. Then "combining" is just adding signed numbers.

Worked Example: $-4x + 9x$

The terms are $-4x$ and $+9x$. Add the coefficients:

$-4 + 9 = 5 \quad\Longrightarrow\quad -4x + 9x = 5x$

Worked Example: $2y - 7y$

The terms are $+2y$ and $-7y$:

$2 - 7 = -5 \quad\Longrightarrow\quad 2y - 7y = -5y$

💡 Reading tip: rewrite subtraction as "adding a negative." $2y - 7y$ becomes $2y + (-7y)$, and now it's a plain addition of $+2$ and $-7$.

Combine Each Pair 🧮

Combine the like terms. Enter the coefficient with its sign (for example, type -5 for $-5x$). Write 0 if the term cancels.

1) $-8x + 3x \;\Rightarrow\; ?\,x$ 2) $6m - 10m \;\Rightarrow\; ?\,m$ 3) $-7y - 4y \;\Rightarrow\; ?\,y$

Pick the Result 🔽

🧩 Combining Like Terms

Part 3 of 5 — Simplifying Longer Expressions

🔑 The Strategy: Real expressions mix several kinds of terms. The plan is always the same: group the like terms, then combine each group. Constants form their own group.

A Reliable 3-Step Method

To simplify an expression like $4x + 3 + 2x + 5$:

1. Identify each term with its sign: $+4x,\; +3,\; +2x,\; +5$.
2. Group like terms together: $(4x + 2x) + (3 + 5)$.
3. Combine each group: $6x + 8$.

$4x + 3 + 2x + 5 = 6x + 8$

⚠️ $6x + 8$ does not simplify further. $6x$ and $8$ are not like terms (one has a variable, one doesn't), so this is the final, fully simplified answer. An answer can have more than one term.

A Worked Example With Subtraction

Simplify $\;9y - 2 - 4y + 7$.

Attach each sign to its term, then group:

$(9y - 4y) + (-2 + 7)$

Combine each group:

$9y - 4y = 5y \qquad -2 + 7 = 5$

$\Longrightarrow\quad 9y - 2 - 4y + 7 = 5y + 5$

💡 A neat habit: list the variable terms first and the constant last, so the answer comes out in the tidy form $5y + 5$.

Group Then Combine 🔽

Simplify $\;7m + 4 - 3m + 6$ one step at a time.

Simplify Fully 🧮

Simplify each expression. Enter the variable-term coefficient in the first box and the constant in the second box. (For $8x + 3$, you'd enter 8 then 3.)

1) $5x + 6 + 2x - 1 = ?\,x + ?$ 2) $10a - 4 - 3a + 9 = ?\,a + ?$

Concept Check 🎯

🧩 Combining Like Terms

Part 4 of 5 — Distributing First, Then Combining

🔑 New Wrinkle: When an expression has parentheses, you must distribute to clear them before you can find like terms. Distribute, then group, then combine.

The Distributive Property

$a(b + c) = ab + ac$

You multiply the outside number by every term inside. Then the parentheses are gone and ordinary combining takes over.

Worked Example: $2(x + 4) + 3x$

Distribute the $2$:

$2(x + 4) = 2x + 8$

Now the expression is $2x + 8 + 3x$. Group and combine:

$(2x + 3x) + 8 = 5x + 8$

⚠️ Distribute to every term inside. A common slip is $2(x+4) = 2x + 4$ — the $4$ got skipped. It must be $2x + 8$.

Concept Check 🎯

Distributing a Negative

A minus sign in front of parentheses means you are multiplying by $-1$ (or a negative number). Every inside sign flips.

Worked Example: $8x - 3(x + 2)$

Distribute the $-3$ across both inside terms:

$-3(x + 2) = -3x - 6$

So the expression becomes $8x - 3x - 6$. Combine:

$8x - 3x - 6 = 5x - 6$

⚠️ The sign trap. $-3(x + 2)$ is $-3x - 6$, not $-3x + 6$. The $-3$ multiplies the $+2$ to give $-6$. Distribute the sign to every term.

Distribute Carefully 🔽

Work through $\;5(y + 3) - 2(y + 1)$.

Distribute, Then Combine 🧮

Simplify to the form $(\text{coefficient})x + (\text{constant})$. Enter the coefficient, then the constant (with its sign).

1) $6(x + 2) - 4x = ?\,x + ?$ 2) $10x - 2(x + 5) = ?\,x + ?$

🧩 Combining Like Terms

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) spot like terms, (2) add their coefficients, (3) simplify long expressions by grouping, and (4) distribute before combining. Let's pull it all together.

Quick Reference

⚠️ Top three traps: (1) $3x + 5x = 8x$, never $8x^2$. (2) $x^2$ and $x$ are not like terms. (3) $-2(x+5) = -2x - 10$, not $-2x + 10$.

Mixed Practice 🎯

One More Round 🧮

Simplify to $(\text{coefficient})x + (\text{constant})$. Enter the coefficient, then the constant (with its sign).

1) $7x + 5 - 7x - 9 \;\Rightarrow\; ?\,x + ?$ (watch the $x$-terms!) 2) $3(2x - 1) + 4 \;\Rightarrow\; ?\,x + ?$

Exit Quiz ✅

Answer all three to finish the lesson.