College Algebra: Key Concepts and Methods Study Guide

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:

• $x + y = 5$

• $x - y = 1$

By adding the equations, $2x = 6 \implies x = 3$. Substitute into the first equation: $3 + y = 5 \implies y = 2$.

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:

• $\begin{bmatrix} 1 & 1 & | & 5 \\ 1 & -1 & | & 1 \end{bmatrix}$

Row operations lead to the solution $x = 3$, $y = 2$.

Complex Numbers

Definition and Operations

Complex numbers extend the real numbers and are written in the form $a + bi$, where $i$ is the imaginary unit ($i^2 = -1$).

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

• Multiplication: Use distributive property and $i^2 = -1$.

• Conjugate: The conjugate of $a + bi$ is $a - bi$.

Example: $(2 + 3i) + (1 - 4i) = 3 - i$ Example: $(2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i^2 = 2 - 5i + 12 = 14 - 5i$

Quadratic Equations

Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. There are several methods to solve them:

• Square Root Property: If $x^2 = k$, then $x = \pm\sqrt{k}$.

• Zero-Factor Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve $x^2 - 5x + 6 = 0$ Factoring: $(x-2)(x-3) = 0 \implies x = 2, 3$

Applications & Modeling with Quadratic Equations

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

• Example: The height $h$ (in meters) of a ball thrown upward is $h = -4.9t^2 + 20t + 1.5$.

• Find when the ball hits the ground: Set $h = 0$ and solve for $t$ 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: $\frac{1}{x} + \frac{2}{x+1} = 3$ Multiply both sides by $x(x+1)$ 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: $\sqrt{x+2} = x - 2$ Square both sides: $x + 2 = (x - 2)^2$ and solve.

Equations with Rational Exponents

Equations may involve exponents that are fractions, such as $x^{1/2}$ or $x^{2/3}$.

• Rewrite using radical notation if helpful.

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

Example: $x^{2/3} = 9$ Raise both sides to the $3/2$ power: $x = 9^{3/2} = (\sqrt{9})^3 = 27$

Equations Quadratic in Form

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

• Let $u = x^n$ to transform the equation into a quadratic in $u$.

Example: $x^4 - 5x^2 + 6 = 0$ Let $u = x^2$, so $u^2 - 5u + 6 = 0$; solve for $u$, then back-substitute for $x$.

Inequalities

Quadratic and Rational Inequalities

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

• Quadratic Inequalities: Solve $ax^2 + bx + c > 0$ by finding roots and testing intervals.

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

Example: Solve $x^2 - 4 < 0$ Roots: $x = -2, 2$. Test intervals: $(-\infty, -2)$, $(-2, 2)$, $(2, \infty)$. Solution: $-2 < x < 2$.

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

Absolute value equations have the form $|A| = B$.

• If $B \geq 0$, then $A = B$ or $A = -B$.

Example: $|x - 3| = 5$ Solutions: $x - 3 = 5 \implies x = 8$ or $x - 3 = -5 \implies x = -2$

Solving Absolute Value Inequalities

• $|A| < B$ (for $B > 0$): $-B < A < B$

• $|A| > B$ (for $B > 0$): $A < -B$ or $A > B$

Example: $|2x + 1| < 3$ $-3 < 2x + 1 < 3 \implies -4 < 2x < 2 \implies -2 < x < 1$

Distance and Midpoint Formulas

Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane is given by:

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Midpoint Formula

The midpoint of the segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is:

• $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Circles: Center-Radius and General Forms

Equations of Circles

A circle with center $(h, k)$ and radius $r$ has the equation:

• $(x - h)^2 + (y - k)^2 = r^2$

The general form is $x^2 + y^2 + Dx + Ey + F = 0$. 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: $f(x)$ denotes the output of function $f$ for input $x$.

Increasing, Decreasing, and Constant Functions

• Increasing: $f(x_1) < f(x_2)$ for $x_1 < x_2$ in an interval.

• Decreasing: $f(x_1) > f(x_2)$ for $x_1 < x_2$ in an interval.

• Constant: $f(x_1) = f(x_2)$ for all $x_1, x_2$ in an interval.

Summary Table: Key Properties of Functions

Property	Description	Example
Domain	All possible input values	$f(x) = \sqrt{x}$, domain: $x \geq 0$
Range	All possible output values	$f(x) = x^2$, range: $y \geq 0$
Increasing	Function rises as $x$ increases	$f(x) = x$ on $(-\infty, \infty)$
Decreasing	Function falls as $x$ increases	$f(x) = -x$ on $(-\infty, \infty)$
Constant	Function remains the same	$f(x) = 3$

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