Solving Linear Equations

Linear Expressions vs. Linear Equations

A linear expression is an algebraic expression of the form ax + b, where a and b are constants. A linear equation is a statement that two linear expressions are equal, typically written as ax + b = c.

• Linear Expression Example:

• Linear Equation Example:

• Solving for a known x: Substitute the value of x and simplify.

• Solving for an unknown x: Find the value(s) of x that make the equation true.

Steps to Solve Linear Equations

1. Distribute constants: Apply the distributive property to remove parentheses.

2. Combine like terms: Add or subtract terms with the same variable.

3. Group terms: Move all terms with x to one side and constants to the other.

4. Isolate x: Solve for x by performing inverse operations.

5. Check solution: Substitute the value of x back into the original equation.

Example

• Equation:

• Step 1: Distribute:

• Step 2: Group terms:

• Step 3: Combine like terms and solve for x.

Linear Equations with Fractions

When linear equations contain fractions, use the Least Common Denominator (LCD) to eliminate fractions before solving.

• Multiply both sides by the LCD to clear denominators.

• Proceed with the standard steps for solving linear equations.

Example

• Equation:

• Step 1: LCD is 12. Multiply both sides by 12 to clear denominators.

• Step 2: Solve the resulting linear equation.

Categorizing Linear Equations

Linear equations can be classified based on the number of solutions:

• Conditional Equation: Has exactly one solution.

• Identity: True for all real numbers (infinitely many solutions).

• Inconsistent Equation: Has no solution.

To determine the type, solve the equation and analyze the result.

Rational Equations

Definition and Solution

A rational equation is an equation that contains one or more rational expressions (fractions with variables in the denominator).

• To solve, multiply both sides by the LCD to eliminate denominators.

• Check for extraneous solutions: Solutions that make any denominator zero are not valid.

Example

• Equation:

• Restriction: (since denominator cannot be zero)

• Step 1: Cross-multiply and solve for x.

• Step 2: Check that the solution does not violate the restriction.

Square Roots of Negative Numbers and Imaginary Unit

Square Roots of Negative Numbers

• The square root of a positive number is real, but the square root of a negative number is not real.

• To handle this, mathematicians defined the imaginary unit such that .

• Any square root of a negative number can be written as , where .

Examples

Powers of the Imaginary Unit

Properties of

• These powers repeat every four exponents.

Shortcut for Evaluating Powers of

• Divide the exponent by 4 and use the remainder to determine the value:

Complex Numbers

Definition and Standard Form

A complex number is a number of the form , where a and b are real numbers and is the imaginary unit.

• Real part:

• Imaginary part:

Examples

• : ,

• : ,

Adding and Subtracting Complex Numbers

• Add or subtract the real parts and the imaginary parts separately.

• Express the answer in standard form .

Example:

Multiplying Complex Numbers

• Use the distributive property (FOIL) as with binomials.

• Replace with and combine like terms.

Example:

Complex Conjugates

• The conjugate of is .

• Multiplying a complex number by its conjugate yields a real number:

Dividing Complex Numbers

• To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator to make the denominator real.

• Expand and simplify to standard form.

Example: Multiply numerator and denominator by .

Summary Table: Operations with Complex Numbers

Key Formulas

• Standard form of a complex number:

• Conjugate:

• Multiplying conjugates:

• Powers of : , , , (repeat every 4)

Additional info: These notes cover foundational Precalculus topics including linear equations, rational equations, and complex numbers, which are essential for further study in algebra, trigonometry, and calculus.