Lesson 7-2: Expand Linear Expressions

Property

A linear expression is an algebraic expression that represents a straight line. To be strictly "linear," every term in the expression must pass these rules:

• Variables (like $x$ or $y$) can only have a visible or invisible exponent of $1$.
• Variables cannot be multiplied together (like $xy$).
• Variables cannot be in the denominator of a fraction (like $\frac{5}{x}$).
• Variables cannot be inside a square root.

Examples

• Linear: $-4x + 7$. (The variable $x$ has an invisible exponent of $1$, and $7$ is a constant).
• Linear: $\frac{1}{2}y - 9$. (The fraction is just a coefficient, the variable $y$ is normal).
• Nonlinear: $5x^2 - 3$. (Fails: The variable is raised to the 2nd power).
• Nonlinear: $\frac{2}{x} + 8$. (Fails: The variable is in the denominator).

Explanation

Why do we care if it's linear? Because linear expressions are the building blocks of Algebra 1! If you see exponents like $2$ or $3$, or variables trapped under fractions, the math rules change completely. When you simplify linear expressions, your final answer should always look like a simple chain of normal variables and constants (e.g., $ax + b$).

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

Two expressions are equivalent if they are both equivalent to a third expression, typically one in the simplified form $ax+b$. To check for equivalence, simplify each expression by applying laws of arithmetic like the distributive property and combining like terms. If they reduce to the same final form, they are equivalent.

Examples

• Are $3(x+4)-5$ and $3x+7$ equivalent? Simplify the first expression: $3x + 12 - 5$ becomes $3x+7$. Since they match, they are equivalent.

• Are $5x+6$ and $2(x+3)+x$ equivalent? Simplify the second expression: $2x+6+x$ becomes $3x+6$. Since $5x+6$ is not the same as $3x+6$, they are not equivalent.