Combining Simple Like Terms

Simplify: $4x + 7x$

Identify like terms: Both $4x$ and $7x$ have the same variable $x$ = Like terms

Add the coefficients: $4 + 7 = 11$ = $11$

Keep the variable: Attach $x$ to the sum = $11x$

Answer: $4x + 7x = 11x$

Subtracting Like Terms

Simplify: $9y - 4y$

Identify like terms: Both $9y$ and $4y$ have the same variable $y$ = Like terms

Subtract the coefficients: $9 - 4 = 5$ = $5$

Keep the variable: Attach $y$ to the result = $5y$

Answer: $9y - 4y = 5y$

Multiple Like Terms

Simplify: $3a + 5b + 4a - 2b$

Group like terms: $3a + 4a$ and $5b - 2b$ = Two groups

Combine the $a$ terms: $3a + 4a = 7a$ = $7a$

Combine the $b$ terms: $5b - 2b = 3b$ = $3b$

Write the final expression: Put the simplified terms together = $7a + 3b$

Answer: $3a + 5b + 4a - 2b = 7a + 3b$

With Constants

Simplify: $2x + 5 + 6x - 3$

Identify all terms: Variable terms: $2x$, $6x$. Constants: $5$, $-3$ = Two types

Combine variable terms: $2x + 6x = 8x$ = $8x$

Combine constants: $5 + (-3) = 5 - 3 = 2$ = $2$

Write the final expression: Variable term + constant = $8x + 2$

Answer: $2x + 5 + 6x - 3 = 8x + 2$

Different Exponents Are Not Like Terms

Simplify: $5x^2 + 3x + 2x^2 - x$

Identify term types: $x^2$ terms: $5x^2$, $2x^2$. $x$ terms: $3x$, $-x$ = Two groups

Combine $x^2$ terms: $5x^2 + 2x^2 = 7x^2$ = $7x^2$

Combine $x$ terms: $3x - x = 3x - 1x = 2x$ = $2x$

Write in standard form: Higher powers first = $7x^2 + 2x$

Answer: $5x^2 + 3x + 2x^2 - x = 7x^2 + 2x$

Mistake: Adding variables: $3x + 5x = 8x^2$

Why: When combining like terms, we add COEFFICIENTS, not exponents. The variable stays the same.

Correct: $3x + 5x = 8x$ (think: 3 apples + 5 apples = 8 apples)

Mistake: Combining unlike terms: $3x + 4y = 7xy$

Why: $x$ and $y$ are different variables, so $3x$ and $4y$ cannot be combined.

Correct: $3x + 4y$ is already simplified (these are unlike terms)

Mistake: Forgetting that $x = 1x$

Why: A variable without a coefficient has an invisible coefficient of 1.

Correct: $x + 5x = 1x + 5x = 6x$

Mistake: Treating $x$ and $x^2$ as like terms

Why: $x$ and $x^2$ have different powers, so they are NOT like terms.

Correct: $x + x^2$ cannot be simplified further

Shopping and Budgets

Combining like items when calculating total costs.

You buy 3 pens at $p$ dollars each, then 5 more pens. Total cost: $3p + 5p = 8p$ dollars.

Sports Statistics

Combining points scored in different games.

A basketball player scores $p$ points in the first half and $2p$ points in the second half. Total: $p + 2p = 3p$ points.

Q: Can I combine $2x$ and $3y$?

A: No! $2x$ and $3y$ have different variables, so they are unlike terms and cannot be combined. $2x + 3y$ is already fully simplified.

Q: Why is $x + x = 2x$ and not $x^2$?

A: Think of it like counting objects: 1 apple + 1 apple = 2 apples, not 1 apple-squared. The exponent stays the same when combining like terms.

Q: What is the coefficient of $-y$?

A: The coefficient is $-1$. When there's no number in front of a variable, it's 1 (or $-1$ if there's a minus sign).