This document provides an overview of solving systems of linear equations with two or three variables. It defines key terms like systems, solutions, and consistent/inconsistent equations. It describes both graphical and algebraic methods. The algebraic methods covered are elimination, substitution, and Cramer's rule. Examples are provided to illustrate each method. Elimination and back substitution are also described as methods for solving three variable systems algebraically.

1. Chapter 6: Systems of Linear Equations Two Variable System

2. Definitions System – a conjunction of two or more equations GENERAL FORM: Solution – an ordered pair of values that will satisfy all equations in the system

3. Solution of Two Variable System Assigning 2 points for each line Computing the intercepts of each line Converting each line into y=mx + b

4. Solution of Two Variable System Isolating one variable Eliminating one variable and solving the other Solving for the determinants

5. Graphical Method Only intersecting equations have a unique or finite solution. They are called consistent equations. However, there are also systems that has infinite solutions or no solutions at all. Inconsistent equations – if two lines are parallel, then the system has no solution

6. Dependent equations – if two lines coincide, then the system has infinite solutions

7. Illustration (1) (2) For equation (1) For equation (2)

13. Algebraic Solution Elimination Method (Gaussian Elimination) Find equations equivalent to the two given equations such that the coefficients of one variable are additive inverse or the same By addition or subtraction of the two equations, we eliminate one variable Derive a linear equation in one variable and solve that variable Substitute the value of the solved variable in the other equation and solve for the second variable

14. Illustration: (1) (2) Multiply (1) by 3 Add (1) and (2) (1) (2)

15. Substitute the value of x to (1) is the solution to the system Thus,

16. Substitution Method Consider the same system: (1) (2) (3) We solve for y in (1) Then we substitute (3) to (2)

17. Substitute the value of x to (3) is the solution to the system Thus,

18. Cramer’s Rule The determinant of the coefficients of the system is Replace the coefficients of x in D by the constant terms to get

19. Similarly, replace the coefficients of y in D by the constants If D ≠ 0 then the solution (x, y) of the system is computed as

20. Using the same system:

21. Exercise Use Elimination, Substitution or Cramer’s Rule to solve for the solution of the system

22. Using Cramer’s Rule:

23. Three Variable System To determine the solution to a system of 3 variables, we can use either Elimination or Cramer’s Rule

24. Elimination: 3 Variable System Equation 1 2 VAR SYSTEM Equation 4 VAR 1 Value of VAR 3 Equation 2 VAR 2 Equation 5 VAR 1 Equation 3 Apply Back Substitution

25. Example 3 Variable System

26. (1) (2) (3)

28. (1) (2) (3)

29. (1)