Linear Systems and Methods of Solution

Solving Linear Systems

Linear systems are collections of two or more linear equations involving the same set of variables. Solving these systems is a fundamental skill in algebra, with applications in science, engineering, and economics.

• Substitution Method: Solve one equation for one variable and substitute this expression into the other equation(s).

• Elimination Method: Add or subtract equations to eliminate one variable, making it possible to solve for the remaining variable(s).

• Special Systems: Systems may have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (dependent).

Example: Solve the system:

By adding the equations, . Substitute into the first equation: .

Matrix Solution of Linear Systems

Matrices provide a systematic way to solve linear systems, especially those with more than two variables.

• Augmented Matrix: Represents the system in matrix form, combining coefficients and constants.

• Gauss-Jordan Elimination: A method to row-reduce the augmented matrix to reduced row-echelon form, revealing the solution.

Example: For the system above, the augmented matrix is:

Row operations lead to the solution , .

Complex Numbers

Definition and Operations

Complex numbers extend the real numbers and are written in the form , where is the imaginary unit ().

• Addition/Subtraction: Combine real and imaginary parts separately.

• Multiplication: Use distributive property and .

• Conjugate: The conjugate of is .

Example: Example:

Quadratic Equations

Solving Quadratic Equations

Quadratic equations are equations of the form . There are several methods to solve them:

• Square Root Property: If , then .

• Zero-Factor Property: If , then or .

• Quadratic Formula:

Example: Solve Factoring:

Applications & Modeling with Quadratic Equations

Quadratic equations model many real-world phenomena, such as projectile motion and area problems.

• Example: The height (in meters) of a ball thrown upward is .

• Find when the ball hits the ground: Set and solve for using the quadratic formula.

Rational and Radical Equations

Solving Rational Equations

Rational equations contain fractions with polynomials in the numerator and denominator.

• Find a common denominator to eliminate fractions.

• Check for extraneous solutions by substituting back into the original equation.

Example: Multiply both sides by and solve.

Solving Radical Equations

Radical equations involve variables under a root.

• Isolate the radical, then raise both sides to the appropriate power.

• Check for extraneous solutions.

Example: Square both sides: and solve.

Equations with Rational Exponents

Equations may involve exponents that are fractions, such as or .

• Rewrite using radical notation if helpful.

• Isolate the term with the rational exponent and raise both sides to the reciprocal power.

Example: Raise both sides to the power:

Equations Quadratic in Form

Some equations, though not quadratic, can be rewritten in quadratic form by substitution.

• Let to transform the equation into a quadratic in .

Example: Let , so ; solve for , then back-substitute for .

Inequalities

Quadratic and Rational Inequalities

Inequalities involve finding the set of values that satisfy a given condition.

• Quadratic Inequalities: Solve by finding roots and testing intervals.

• Rational Inequalities: Set the rational expression to zero, find critical points, and test intervals.

Example: Solve Roots: . Test intervals: , , . Solution: .

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

Absolute value equations have the form .

• If , then or .

Example: Solutions: or

Solving Absolute Value Inequalities

• (for ):

• (for ): or

Example:

Distance and Midpoint Formulas

Distance Formula

The distance between two points and in the plane is given by:

Midpoint Formula

The midpoint of the segment joining and is:

Circles: Center-Radius and General Forms

Equations of Circles

A circle with center and radius has the equation:

The general form is . Completing the square can convert the general form to center-radius form.

Functions and Relations

Definitions and Notation

A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Domain: The set of all possible input values.

• Range: The set of all possible output values.

• Function Notation: denotes the output of function for input .

Increasing, Decreasing, and Constant Functions

• Increasing: for in an interval.

• Decreasing: for in an interval.

• Constant: for all in an interval.

Summary Table: Key Properties of Functions

Additional info: Some explanations and examples were expanded for clarity and completeness based on standard College Algebra curriculum.