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:

• Slope-Intercept Form:

• Point-Slope Form:

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

Example: Solve .

• Subtract 3 from both sides:

• Divide by 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:

Solution:

• Add equations:

• Substitute into first equation:

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: can often be factored into .

• Difference of Squares:

• Perfect Square Trinomial:

Example: Factor .

Solving Quadratic Equations

Methods for Solving Quadratics

Quadratic equations are equations of the form . They can be solved using several methods:

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

• Quadratic Formula:

• Completing the Square: Rewrite the equation in the form and solve for .

Example: Solve by factoring.

• So, or

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:

  • (for )

Example: Simplify .

• Factor numerator:

• Factor denominator:

• Simplify: (for )

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 .

• Square both sides:

Solving Inequalities

Linear and Nonlinear Inequalities

Inequalities compare expressions using symbols such as . 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 .

• Add 3:

• Divide by 2:

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 be the number.