Lesson 7-1: Combine Like Terms

Property

To work with algebra, you need to speak the language. An expression is made up of separate parts called terms (separated by $+$ or $-$ signs).

• Coefficient: The number physically attached to the front of a variable (it multiplies the variable). If a variable stands alone like $x$, its coefficient is an invisible $1$.
• Constant: A plain number with no variable attached.
• Like Terms: Terms that have the exact same variable(s) raised to the exact same power. Constants are always like terms with other constants.

Examples

• Anatomy: In the expression $5x - 3y + 8 - x$:
  • There are 4 terms.
  • The coefficients are $5$, $-3$, and $-1$ (from the $-x$).
  • The constant is $8$.
• Identifying Like Terms: $4x$ and $-9x$ are like terms. $7x$ and $7y$ are NOT. $3x$ and $3x^2$ are NOT (the exponents are different). $5$ and $-12$ ARE like terms.

Explanation

Think of like terms as specific categories of items. Variables act like unit labels. You can add 3 Apples and 4 Apples to get 7 Apples ($3a + 4a = 7a$). But you cannot mathematically combine 3 Apples and 4 Bananas ($3a + 4b$). They just sit next to each other in the expression.

Property

To simplify an expression means to combine all possible like terms into a single term.

1. Group the like terms together (mentally or physically).
2. Add or subtract their coefficients.
3. Never change the variable or the exponent when adding or subtracting!

Crucial Rule: Always "capture" the sign ($+$ or $-$) immediately to the left of a term. The sign belongs to that term. Treating subtraction as "adding a negative" prevents most errors.

Examples

• Basic Grouping: Simplify $7b + 5 - 3b + 2$.
  • Combine the $b$'s: $7b - 3b = 4b$.
  • Combine the constants: $5 + 2 = 7$.
  • Final Answer: $4b + 7$.
• Capturing the Sign: Simplify $4x - 8y - x + 3y$.
  • The terms are: $4x$, $-8y$, $-1x$, $+3y$.
  • Combine the $x$'s: $4x - 1x = 3x$.
  • Combine the $y$'s: $-8y + 3y = -5y$.
  • Final Answer: $3x - 5y$.

Explanation

The most common mistake students make is dropping or confusing negative signs. When you see $5x - 8x$, do not think of it as "five minus eight." Think of it as combining a positive $5$ and a negative $8$, which results in a negative $3$ ($-3x$). Circling or underlining like terms (along with the $+$ or $-$ sign right in front of them) is the best way to keep your math clean and accurate.

Property

In real-world geometry or word problems, you often need to find a "Total" (like the perimeter of a shape or the total money spent). You create an algebraic model by writing an expression that adds all the individual parts together, and then you simplify it by combining like terms.

Examples

• Finding Perimeter: A triangle has three side lengths given as algebraic expressions: $(3x)$, $(x + 4)$, and $(2x - 1)$.
  • Step 1 (Write the sum): $P = 3x + x + 4 + 2x - 1$
  • Step 2 (Combine like terms): $P = (3x + 1x + 2x) + (4 - 1)$
  • Final Simplified Perimeter: $P = 6x + 3$.
• Finding Totals: You buy $3$ shirts that cost $d$ dollars each, and a pair of shoes that costs $40$. Your friend buys $2$ shirts ($d$ dollars each) and a hat for $15$.
  • Your cost: $3d + 40$. Friend's cost: $2d + 15$.
  • Total cost expression: $(3d + 40) + (2d + 15) = 5d + 55$.

Explanation

Real-world problems rarely give you a neat equation right away. Your job is to translate the physical situation into algebra. Remember that finding a perimeter always means adding up all the outside edges. If a side doesn't have a number in front of the variable (like just "$x$"), never forget to count it as $1x$ when you are adding up your total!