Chapter 2
Systems of Equations
& Inequalities
This chapter consists of 3 sections as follows:
2.1 System of Linear Equations in Two Variables
2.2 System of Linear Equations in more than Two Variables(Three Variables)
2.3 Inequalities
System of Linear 2.1
Equations in Two
Variables
1. Systems of Linear Equations
in Two Variables
A linear equation in two variables takes the form
. 8 2
, 6 2
y x
y x
, C By
Ax B
A ,
Where are not both zero
Any collection of two or more equations is called a
system of equations. For example, the system of
two linear equations in two variables is a set as
Independent, inconsistent and dependent equations
Independent system
solution One
Inconsistent system Dependent system solution No
Many solution
y y y
x x x
Lines have different slopes
Lines have the
same slopes Lines are the
same
Methods for Solving
System of Linear Equations
I. Substitution Method
Solution
Example 1
Solve
400 06
. 0 05
. 0 .
23 5
6
, 1000 2
) ,
6 3
)
y x
y x
x y
b y
x a
6 3
) x y
a y 3 x 6 ( 1 )
23 30
15
6
x x
equation second
in the
Substitute y 6 x 5 ( 3 x 6 ) 23 7
21
x 3
1
x (1)
3 in substitute 1
find
To y x 6 5
3
3 1
y
equations original
both the satisfies
solution the
Check that
) 5 3 ,
( 1 point the
is solution
The
) 1 ( 1000
2
) y x b
400 60
12 . 0 05
.
0
x x
equation second
in the Substitute y
400 )
1000 2
( 06 . 0 05
.
0
x x
2000
x (1) in 2000 substitute
find
To y x
2000 1000 5000
2
y
) 5000 ,
2000 (
point the
is
solution
The
Class Work
Solve the following systems of equations by substitution
. 1 2
, 9 3
)
y x
y x
i
. 8 2
, 6 2
)
y x
y x
ii
Exercise Page(798-799) 11
In Exercises 1 to 20,solve each system of equations by the substitution method .
5 3y
- 3x 4
5
1 3
3 11. 4
4 3
1 5
7. 6
5 2
6 3
5. 2
2 x
16 3y
- 1. 2x
x y
x y
x y
y x
y x
y x
y
x
II. Elimination Method
Solution
Example 3
Solve
12 2
3 .
1 2
, 2 3
2 ) ,
9 3
)
y x
y x
y x
b y
x a
10 5
1 2
9 3
)
x
y x
y x
a
1 )
2 (
2
y
2
x
3
y
y y
x
x 2 in 2 1 to find use
Now
) 3 , 2 ( point the
is solution
The
30 5
24 4
6
6 9
6
y y x
y x
2 )
6 ( 3
2
x
6
y
16
2
x
x y
x
y 6 in first eqn. 2 3 2 to find use
Now
equations original
both the satisfies
solution the
Check that
) 6 , 8 ( point the
is solution The
: 2 - by second the
and 3 by eqn first he
multiply t ,
eliminate
b)To x
8
x
Class Work
Find x, y by elimination method
. 1 5
3
, 5 3
4 )
y x
y x
i
. 9 5
2
, 8 7
5 )
y x
y x
ii
Exercise Page(798-799) 12
In Exercises 21 to 40,solve each system of equations by the elimination method .
9 4
2
11 6
29. 3
0
6 10
0 3
25. 5
12 4
5
21 7
23. 4
4
3 4
10 21. 3
y x
y x
y x
y x
y x
y x
y x
y
x
