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

2 3 6

6 4 0

x y

x y

ì + =ïïíï - =ïî

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 systemClass 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 )= 6 + 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 = 8

- 4 x – 2 y = - 16

Solution Set = Infinite = {(x,y)/ 2x + y = 8 }

Dependent