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Chapter 5

Chapter 5. Section 5.1 System of Linear Equations in Two Variables. 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. Some Definitions and Illustrations.

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Chapter 5

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  1. Chapter 5 Section 5.1 System of Linear Equations in Two Variables 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

  2. 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 Class Work Give 3 examples of a linear system

  3. Objective 11. 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: Equation 2 (a) Equation 1 Substitute x = 3 and y = 2 in the equation 2x + y = 8 Substitute x = 3 and y = 2 in the equation x – 2y = -1 L.S : 2 ( 3) + ( 2 ) = 6 + 2 = 8 L.S : ( 3) – 2 ( 2 ) = 3 – 4 = -1 R.S : 8 Same answer for equation 2 R.S : - 1 Same answer for equation 1 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

  4. Objective 22. 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 2: Solve the linear system by graphing 2x + y = 4 4x + 2y = 8 Example 2: Solve the linear system by graphing 2x + y = 4 2x + y = 5 Example 1: Solve the linear system by graphing 2x + y = 4 x – y = 5 (3 , - 2 ) Solution Set = { ( 3 , - 2 ) } Consistent System Solution Set = { } = Ø Inconsistent System Solution Set = Infinite = { (x , y ) / 2x + y = 4 } Dependent System

  5. 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 Solution Set = { ( 5 , - 3 ) } Consistent i) 3x – y = 4 3x - y = 1 Solution Set = { } = Ø Inconsistent iii) 3x + 2 y = 12 y = 3 Solution Set = { ( 2 , 3 ) } consistent

  6. iv) 2x + y = 8 - 4 x – 2 y = - 16 Solution Set = Infinite = {(x,y)/ 2x + y = 8 } Dependent

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