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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
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
ìïï 2 x + 3 y = 6
í
ïïî 6 x - 4 y = 0
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
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
L.S :
R.S :
Equation 2
Substitute x = 3 and y = 2 in the equation
( 3) – 2 ( 2 ) = 3 – 4 = -1
-1
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
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
Equation 2
L.S = - 2
and
R.S = 8
Satisfies equation 1
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
Type 3: Infinite Number of Solution points
point
( Dependent )
( Inconsistent )
Example 1: Solve the linear
system by graphing
2x + y = 4
x–y=5
Example 2: Solve the linear
system by graphing
Example 2: Solve the linear
system by graphing
2x + y = 4
2x + y = 4
2x + y = 5
4x + 2y = 8
(3 , - 2 )
Solution Set = { ( 3 , - 2 ) }
Consistent System
Solution Set = {
} =Ø
Inconsistent System
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
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
iv)
2x +
y=8
- 4 x – 2 y = - 16
Solution Set = Infinite = {(x,y)/ 2x + y = 8 }
Dependent