Core Concepts in College Algebra: Equations, Factoring, and Problem Solving

Linear Equations and Models

Understanding Linear Equations

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. They are fundamental in modeling real-world relationships where the rate of change is constant.

• Standard Form: $Ax + By = C$

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Solution: The value(s) of the variable(s) that make the equation true.

Example: Solve $2x + 3 = 7$.

• Subtract 3 from both sides: $2x = 4$

• Divide by 2: $x = 2$

Solving Systems of Linear Equations

Methods for Solving Systems

A system of linear equations consists of two or more linear equations with the same variables. The solution is the set of variable values that satisfy all equations simultaneously.

• Graphical Method: Plot each equation and identify the intersection point(s).

• Substitution Method: Solve one equation for a variable and substitute into the other.

• Elimination Method: Add or subtract equations to eliminate a variable.

Example: Solve the system:

• $x + y = 5$

• $x - y = 1$

Solution:

• Add equations: $2x = 6 \Rightarrow x = 3$

• Substitute into first equation: $3 + y = 5 \Rightarrow y = 2$

Factoring

Factoring Algebraic Expressions

Factoring is the process of writing an expression as a product of its factors. It is essential for simplifying expressions and solving equations.

• Greatest Common Factor (GCF): Factor out the largest common factor from all terms.

• Factoring Trinomials: $ax^2 + bx + c$ can often be factored into $(mx + n)(px + q)$.

• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

• Perfect Square Trinomial: $a^2 + 2ab + b^2 = (a + b)^2$

Example: Factor $x^2 - 9$.

• $x^2 - 9 = (x + 3)(x - 3)$

Solving Quadratic Equations

Methods for Solving Quadratics

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved using several methods:

• Factoring: Express as a product of binomials and set each factor to zero.

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

• Completing the Square: Rewrite the equation in the form $(x + p)^2 = q$ and solve for $x$.

Example: Solve $x^2 - 5x + 6 = 0$ by factoring.

• $(x - 2)(x - 3) = 0$

• So, $x = 2$ or $x = 3$

Rational Expressions and Exponents

Working with Rational Expressions and Exponents

Rational expressions are fractions where the numerator and/or denominator are polynomials. Exponents indicate repeated multiplication.

• Simplifying Rational Expressions: Factor numerator and denominator, then reduce common factors.

• Properties of Exponents:

  • $a^m \cdot a^n = a^{m+n}$

  • $\frac{a^m}{a^n} = a^{m-n}$

  • $(a^m)^n = a^{mn}$

  • $a^0 = 1$ (for $a \neq 0$)

  • $a^{-n} = \frac{1}{a^n}$

Example: Simplify $\frac{x^2 - 9}{x^2 - 6x + 9}$.

• Factor numerator: $x^2 - 9 = (x + 3)(x - 3)$

• Factor denominator: $x^2 - 6x + 9 = (x - 3)^2$

• Simplify: $\frac{(x + 3)(x - 3)}{(x - 3)(x - 3)} = \frac{x + 3}{x - 3}$ (for $x \neq 3$)

Solving Radical and Factorable Equations

Equations Involving Radicals and Factoring

Radical equations contain variables within a root. Factorable equations can be solved by expressing them as products of factors.

• Solving Radical Equations: Isolate the radical, then raise both sides to the appropriate power to eliminate the radical.

• Check for Extraneous Solutions: Always substitute solutions back into the original equation to verify.

• Solving by Factoring: Set the equation to zero, factor, and set each factor equal to zero.

Example: Solve $\sqrt{x + 3} = 5$.

• Square both sides: $x + 3 = 25$

• $x = 22$

Solving Inequalities

Linear and Nonlinear Inequalities

Inequalities compare expressions using symbols such as $<, >, \leq, \geq$. Solutions are often represented on a number line or in interval notation.

• Solving Linear Inequalities: Use similar steps as equations, but reverse the inequality sign when multiplying or dividing by a negative number.

• Compound Inequalities: Combine two inequalities using 'and' or 'or'.

• Quadratic and Rational Inequalities: Set the inequality to zero, factor, and test intervals between critical points.

Example: Solve $2x - 3 < 7$.

• Add 3: $2x < 10$

• Divide by 2: $x < 5$

Problem Solving

Applying Algebraic Methods to Real-World Problems

Problem solving in algebra involves translating word problems into mathematical equations, solving, and interpreting the results in context.

• Identify Variables: Assign variables to unknown quantities.

• Set Up Equations: Translate the problem statement into one or more equations.

• Solve and Interpret: Solve the equations and interpret the solution in the context of the problem.

Example: A number increased by 7 is 15. What is the number?

• Let $x$ be the number. $x + 7 = 15$

• $x = 8$