This section details two primary methods for solving systems of equations: substitution and comparison methods.

Definition: The substitution method involves expressing one variable in terms of another and substituting it into the remaining equation.

Example: For the system: 2x + y = 3 x + 2y = 9 Solve for y in terms of x: y = 3 - 2x Then substitute into the second equation.

Highlight: The comparison method involves setting equal two different expressions for the same variable.

This page explains the addition and subtraction method for solving systems of equations.

Definition: This method involves manipulating equations to eliminate one variable by adding or subtracting equations.

Example: For the system: 2x - 3y = 58 5x + 3y = 90 Add equations to eliminate y variable.

Highlight: The equations must be in standard form before applying this method.

This section introduces Cramer's Rule as a systematic method for solving systems of equations.

Definition: Cramer's Rule uses determinants to find solutions to systems of equations.

Vocabulary:

• Determinant (D): The value ab' - a'b in a 2x2 system
• Dx: Determinant for x-variable
• Dy: Determinant for y-variable

Highlight: The system's nature can be determined by analyzing the determinants:

• If D ≠ 0: System is determined
• If D = 0: System is either impossible or indeterminate

This page covers special cases in systems of equations and their analysis.

Definition: Special cases occur when coefficient ratios have specific relationships.

Example: When the ratio of x-coefficients equals the ratio of y-coefficients and terms, the system is indeterminate.

Highlight: Understanding coefficient relationships helps predict system type before solving.

This section focuses on solving systems involving fractions.

Definition: Fractional systems are those where equations contain fractions with variables.

Highlight: Key steps for solving fractional systems:

1. Find common denominators
2. Multiply all terms to eliminate fractions
3. Solve resulting system using standard methods

This final section extends concepts to systems with three variables.

Definition: Three-variable systems require finding an ordered triple of numbers that satisfies all equations.

Example: Solving methods include:

• Substitution
• Addition/Subtraction
• Sequential elimination of variables

Highlight: Three-variable systems require more systematic approach and careful organization of steps.

This page introduces fundamental concepts of systems of equations and their classifications. The content focuses on systems with two equations and two unknowns.

Definition: A system of equations is a set of two or more equations with the same unknowns that must be satisfied simultaneously.

Vocabulary:

• Determined system: Has a finite number of solutions
• Impossible system: Has no solutions
• Indeterminate system: Has infinite solutions

Example: A determined system can be visualized as two lines intersecting at a single point, while parallel lines represent an impossible system.

Highlight: The degree of a system is determined by the product of the degrees of its equations.