Combining Two Like Terms

Simplify: $7x + 4x$

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

Add the coefficients: $7 + 4 = 11$ = New coefficient is $11$

Write the simplified expression: $7x + 4x = 11x$ = $11x$

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

Combining Multiple Like Terms

Simplify: $5a + 3b - 2a + 6b$

Identify the like terms: Terms with $a$: $5a$ and $-2a$ Terms with $b$: $3b$ and $6b$ = Two groups of like terms

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

Combine the $b$ terms: $3b + 6b = 9b$ = $9b$

Write the simplified expression: $5a + 3b - 2a + 6b = 3a + 9b$ = $3a + 9b$

Answer: $5a + 3b - 2a + 6b = 3a + 9b$

Including Constants

Simplify: $4x + 7 - 2x + 3$

Identify all like terms: Variable terms: $4x$ and $-2x$ Constant terms: $7$ and $3$ = Two groups of like terms

Combine the variable terms: $4x - 2x = 2x$ = $2x$

Combine the constants: $7 + 3 = 10$ = $10$

Write the simplified expression: $4x + 7 - 2x + 3 = 2x + 10$ = $2x + 10$

Answer: $4x + 7 - 2x + 3 = 2x + 10$

Working with Subtraction

Simplify: $8y - 3y - 5y$

Verify all terms are like terms: All three terms have the variable $y$ = All are like terms

Combine from left to right: $8y - 3y = 5y$ $5y - 5y = 0$ = $0$

Write the final answer: When all variable terms cancel, the result is $0$ = $0$

Answer: $8y - 3y - 5y = 0$

Multiple Variables and Powers

Simplify: $3x^2 + 2x - x^2 + 5x - 4$

Group like terms: $x^2$ terms: $3x^2$ and $-x^2$ $x$ terms: $2x$ and $5x$ Constants: $-4$ = Three groups

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

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

Bring down the constant: The constant $-4$ stays as is = $-4$

Write in standard form: Highest power first: $2x^2 + 7x - 4$ = $2x^2 + 7x - 4$

Answer: $3x^2 + 2x - x^2 + 5x - 4 = 2x^2 + 7x - 4$

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

Why: Different variables cannot be combined. $x$ and $y$ represent different unknown values.

Correct: $3x + 4y$ is already simplified - these terms cannot be combined.

Mistake: Adding different powers: $2x + 3x^2 = 5x^2$ or $5x^3$

Why: $x$ and $x^2$ are not like terms because they have different exponents.

Correct: $2x + 3x^2$ is already simplified. Write in standard form: $3x^2 + 2x$

Mistake: Forgetting negative signs: $5x - 3x = 8x$

Why: Subtraction means adding a negative: $5x - 3x = 5x + (-3x)$

Correct: $5x - 3x = 2x$ (we subtract 3 from 5)

Mistake: Multiplying instead of adding coefficients: $4x + 2x = 8x$

Why: When combining like terms, we add (or subtract) the coefficients, not multiply them.

Correct: $4x + 2x = 6x$ (add 4 and 2 to get 6)

Shopping Calculations

When buying multiple items, simplifying helps find the total cost quickly.

If shirts cost $x$ dollars each, buying 3 shirts and then 2 more shirts costs $3x + 2x = 5x$ dollars total.

Perimeter of Shapes

Finding perimeters often requires simplifying expressions with variables.

A rectangle has length $2x$ and width $x$. Perimeter = $2x + x + 2x + x = 6x$

Q: Why can't I combine $3x$ and $5y$?

A: Because $x$ and $y$ represent different unknown values. $3x$ means '3 times some number x' and $5y$ means '5 times some different number y'. Without knowing what $x$ and $y$ are, we can't combine them into one term.

Q: Is $x$ the same as $1x$?

A: Yes! When a variable has no visible coefficient, the coefficient is 1. So $x = 1x$, and when simplifying, we count $x$ as having coefficient 1.

Q: Why is $x^2$ different from $x$?

A: $x^2$ means '$x$ times $x$' while $x$ is just '$x$'. They represent different quantities. For example, if $x = 3$, then $x = 3$ but $x^2 = 9$. That's why $2x + 3x^2$ cannot be simplified further.

Q: What if all terms cancel out?

A: If all terms cancel, the answer is $0$. For example, $5x - 5x = 0$. This is a valid simplified result!