1.1 Linear Equations

Definition and Overview

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. The general form of a linear equation in one variable is:

• Standard Form: , where a and b are constants and a \neq 0.

Solving a linear equation involves isolating the variable on one side of the equation.

Solving Simple Linear Equations

• Step 1: Simplify both sides of the equation if necessary (combine like terms, remove parentheses).

• Step 2: Use addition or subtraction to isolate terms containing the variable on one side.

• Step 3: Use multiplication or division to solve for the variable.

Example 1:

• Subtract 3 from both sides:

• Divide both sides by 2:

Example 2:

• Add 5 to both sides:

Solving Linear Equations with Fractions

• Clear denominators by multiplying both sides by the least common denominator (LCD).

• Proceed as with standard linear equations.

Example:

• Combine like terms:

• Subtract from both sides: (No solution in this case)

Solving Linear Equations with Variables on Both Sides

• Move all variable terms to one side and constants to the other.

• Simplify and solve for the variable.

Example:

• Combine like terms:

• Subtract from both sides:

• Divide by 14:

1.6 Quadratic Equations

Definition and Overview

A quadratic equation is a second-degree polynomial equation in one variable, generally written as:

• Standard Form: , where a \neq 0.

Solving Quadratic Equations by Factoring

• Rewrite the equation in standard form.

• Factor the quadratic expression if possible.

• Set each factor equal to zero and solve for the variable.

Example:

• Factor:

• Set each factor to zero: or

• Solutions: or

Solving Quadratic Equations by the Quadratic Formula

• If factoring is not possible, use the quadratic formula:

• Where , , and are coefficients from .

Example:

• Here, , ,

• Plug into the formula:

• So or

Solving Quadratic Equations by Completing the Square

• Move constant term to the other side.

• Add the square of half the coefficient of to both sides.

• Write the left side as a squared binomial and solve for .

Example:

• Move 1:

• Add to both sides:

• Write as

• Take square root:

• So

Key Properties and Tips

• Always check for extraneous solutions, especially when dealing with rational equations.

• Quadratic equations can have two real solutions, one real solution, or two complex solutions depending on the discriminant .

• Linear equations always have one solution unless the equation is inconsistent (no solution) or dependent (infinitely many solutions).

Summary Table: Methods for Solving Equations