Linear Notes

1. Definition of Linear Expressions

A linear expression is an algebraic expression in which the highest power (exponent) of the variable is $1$. It typically takes the form:
$ax + b$
Where:

• $a$ is the coefficient of the variable $x$.

• $x$ is the variable.

• $b$ is the constant.

2. Components of Linear Expressions

• Variables: Symbols (usually letters) used to represent unknown values (e.g., $x, y, z$).

• Coefficients: The numerical factor multiplied by a variable (e.g., in $4y$, $4$ is the coefficient).

• Constants: A fixed numerical value that does not change because it is not attached to a variable (e.g., in $7x - 3$, $-3$ is the constant).

• Terms: Parts of the expression separated by plus ($+$) or minus ($-$) signs (e.g., $5x$ and $8$ are the terms in $5x + 8$).

3. Key Characteristics

• The degree of a linear expression is always $1$.

• The variable cannot be in the denominator (e.g., $\frac{1}{x}$ is not linear).

• There are no variables under radical signs (e.g., $\sqrt{x}$ is not linear).

• There are no products of variables (e.g., $xy$ is not linear).

4. Simplifying Linear Expressions

Simplifying involves combining like terms, which are terms with the exact same variable and exponent.

• Steps to Simplify:

  1. Identify like terms (terms with the same variable).

  2. Identify constants.

  3. Use the distributive property if there are parentheses: $a(b + c) = ab + ac$.

  4. Combine the coefficients of like terms.

• Example: Simplify $4x + 7 - 2x + 5$

  • Group like terms: $(4x - 2x) + (7 + 5)$

  • Combine: $2x + 12$

5. Evaluating Linear Expressions

Evaluating an expression means finding its numerical value by substituting a given number for the variable.

• Example: Evaluate $3x - 4$ for $x = 5$

  1. Substitute: $3(5) - 4$

  2. Multiply: $15 - 4$

  3. Subtract: $11$

6. Common Examples

• Linear Expressions:

  • $x + 10$

  • $\frac{1}{2}y - 4$

  • $-3z$

• Non-Linear Expressions:

  • $x^2 + 5$ (Degree is 2)

  • $\frac{5}{x}$ (Variable in denominator)

  • $3\sqrt{x}$ (Variable under radical)