Chapter 5 Objectives 1. Find ordered pairs associated with two equations 2. Solve a system by graphing 3. Solve a system by the addition method 4. Solve a system by the substitution method Section 5.1 System of Linear Equations in Two Variables
Some Definitions and Illustrations System of equations: Whenever two or more equations are combined together, they form a system Example 1. 2x + 3y = 6 is an equation 6x – 4y = 0 is also an equation. Now combing both equations together gives a system is a system of equations the equations are linear; therefore, the system is called a linear system Give 3 examples of a linear system Class Work
Objective 1 1. Find ordered pairs associated with two equations Definition: A solution for a linear system of equations in two variables is an ordered pair of real numbers (x,y) that satisfies both equations in the system. Example 2: Given the linear system x – 2y = -1 2x + y = 8 (a) Check if the ordered pair ( 3, 2 ) is a solution to the system (b) Check if the ordered pair ( -1, 0 ) is a solution to the system Solution: (a) Equation 1 x – 2y = -1 Substitute x = 3 and y = 2 in the equation L.S : ( 3) – 2 ( 2 ) = 3 – 4 = -1 R.S : - 1 Same answer for equation 1 Equation 2 2x + y = 8 Substitute x = 3 and y = 2 in the equation L.S : 2 ( 3) + ( 2 ) = = 8 R.S : 8 Same answer for equation 2 Answer: The ordered pair satisfies both equations, therefore, it is a solution point for the linear system. The solution Set = { ( 3, 2 ) } (b) Let’s do it as class work Answer: Equation 1 L.S = -1 and R.S = -1 Satisfies equation 1 Equation 2 L.S = - 2 and R.S = 8 DOES NOT satisfy equation 2. Conclusion: The ordered pair ( -1, 0 ) is not a solution point for the linear system
Objective 2 2. Solve a system by graphing Type 1: Only One Solution point ( Consistent ) Type 2: NO Solution point ( Inconsistent ) Type 3: Infinite Number of Solution points ( Dependent ) Example 1: Solve the linear system by graphing 2x + y = 4 x – y = 5 (3, - 2 ) Solution Set = { ( 3, - 2 ) } Consistent System Example 2: Solve the linear system by graphing 2x + y = 4 2x + y = 5 Solution Set = { } = Ø Inconsistent System Example 2: Solve the linear system by graphing 2x + y = 4 4x + 2y = 8 Solution Set = Infinite = { (x, y ) / 2x + y = 4 } Dependent System
Class Work Example 1: Solve the linear systems by graphing “Use Derive “ and identify if the system is consistent, inconsistent or dependent i) x – y = 8 x + y = 2 i) 3x – y = 4 3x - y = 1 iii) 3x + 2 y = 12 y = 3 Solution Set = { ( 5, - 3 ) } Consistent Solution Set = { } = Ø Inconsistent Solution Set = { ( 2, 3 ) } consistent
iv) 2x + y = x – 2 y = - 16 Solution Set = Infinite = { (x,y)/ 2x + y = 8 } Dependent