Lesson 2: Adding and Subtracting Linear Expressions

Property

Linear expressions can be added or subtracted using two organizational methods:

• Vertical Method: Align like terms in columns and combine vertically
• Horizontal Method: Use parentheses and properties of operations to group and combine like terms

Examples

Property

To expand an expression means to remove the parentheses. We do this using the Distributive Property: $a(bx + c) = abx + ac$. You must multiply the outside number by every single term inside the parentheses. After expanding, you finish the job by combining any like terms.

Examples

• Basic Expansion: Expand $3(2x + 5)$.
  • Distribute: $3 \cdot 2x + 3 \cdot 5 = 6x + 15$.
• Expand and Combine: Simplify $4(x - 8) - x$.
  • Distribute the 4: $4x - 32 - x$.
  • Combine like terms ($4x$ and $-x$): $3x - 32$.
• The Negative Ninja (Trap): Expand $-2(4x - 7)$.
  • Distribute $-2$ to $4x$: $-8x$.
  • Distribute $-2$ to $-7$: $+14$ (Negative times Negative is Positive!).
  • Answer: $-8x + 14$.

Explanation

There are two massive traps when expanding expressions.
Trap 1: "Dropping a term." Students often multiply the outside number by the first term, but forget to multiply it by the second term! (e.g., writing $3(x+4)$ as $3x+4$ instead of $3x+12$).
Trap 2: "The Ninja Negative." If there is a negative sign outside the parenthesis, like $-(x - 3)$, it acts as a $-1$. It sneaks in and flips the sign of EVERY term inside. It becomes $-x + 3$. Stay alert!

Property

Factoring is the process of using the distributive property in reverse.
To factor an expression, find the greatest common factor (GCF) of the terms, write it outside the parentheses, and then determine what remains inside the parentheses.

This can be modeled by arranging algebra tiles into a rectangle and finding the dimensions.

Examples

• To factor $6x + 18$, find the greatest common factor of $6x$ and $18$, which is $6$. Write $6$ outside parentheses. This gives $6(x + 3)$.
• To factor $5y^2 - 10y$, the greatest common factor is $5y$. Factoring this out gives $5y(y - 2)$.
• To factor $8a + 12b$, the greatest common factor of $8a$ and $12b$ is $4$. The factored expression is $4(2a + 3b)$.

Explanation

Factoring is like being a math detective. You start with the final expression and work backward to find the original factors that were multiplied together. It's the opposite of distributing; you're 'un-distributing' the expression.