Chapter 5.1 Systems of Equations in

Two Unknowns

Linear Systems in Two Unknowns

1 1 1

2 2 2

linear system of two equations in

two vari

A

is of the able f ms or

a x b y c

a x b y c

Nonlinear Systems in Two Unknowns

2 2

has at least one nonlinear

equation.

2 11

500

1

Nonlinear system of two equations in

two unknow

2 4 2

ns

1004

x yx y

x y

x y

Solutions to Systems of Equations

A

is an ordered pair that makes both

equations true.

Geometrically,

solution of a system

it is where the graph

of two equations in two

variables

s of the two

equtions intersect.

Classification of System of Equations

A system is said to be:

1. if it has a solution.

a. if it has finite

number of solutions.

b. if it has infinitely

many solutions.

2

consistent

independent

dependent

inc. if it has no son olsistent ution.

Solving Systems of Equations

Methods for solving system of equations:

1. Substitution

2. Elimination

3. Using Matrices

Example 5.1.1

Solve the following system of equations using

the method of substitution.

2 3 38 11.

16 2

16 substitute 4 in 3 :

16 3 16 6

10

substitute 3 in 1 :

2 16 3 38 10,6

32 2 3 38

6 4

x y

x y

x y

x y x

x

y y SS

y y

y

1 1 75

675 5

61 1 75 5

5 56 7

5 56 7

2 11

2.1 2

4

Let and 4

2 1 1Substitute 4 in 3 :

2 4 2

4 2 3 4 2

substitute 3 in 1 :

2 4 2 1

8 4 1 ,

5 7

x y

x y

x y

x y

u v v

u v

u v

u v u

v v x y

v v SS

v

2 2

22

2 2

2

2

2

500 13.

4 2 100 2

2 100 4

50 2 3

Substitute 3 in 1 : Substitute 4 in 3 :

50 2 500 50 2 20 10

2500 200 4 500

5 200 2000 0 20,10

40 400 0

20 0

20 0

20 4

x y

x y

y x

y x

x x y

x x x

x x SS

x x

x

x

x

2 2 50020,10

4 2 100

x ySS

x y

-50 -40 -30 -20 -10 10 20 30 40 50

-50

-40

-30

-20

-10

10

20

30

40

50

x

y

20,10

Example 5.1.2

Solve the following system of equations using

the method of elimination.

2 3 38 11.

16 2

Eliminate :

2 3 38

2 2 32

6 3

Substitute 3 in 2 :

6 16

10 10,6

x y

x y

x

x y

x y

y

x

x SS

423

423

823

1523

5 5 423 23 23

3 2 1 12.

4 5 0 2

Eliminate :

12 8 4

12 15 0

23 4

3

Substitute 3 in 1 :

3 2 1

3 1

3

,

x y

x y

x

x y

x y

y

y

x

x

x

x SS

2 203.

3 6 60

Elimintate :

3 6 60

3 6 60

0 0

The system has infinitely many solutions.

The system is dependent.

2 20

20

2

20,

2

x y

x y

y

x y

x y

y x

xy

xSS x y y

2 2

2 2

2

22 2 2

2 2 2

2 2

2

2 14.

4 25

Eliminate : 2 :

2 1 2 2 1

4 25 8 1

6 24 9

4 3

2

x y

x y

x y

x y x

x y x

y x

y x

y

2 2

2 2

22

2

2

2 12: 3

4 25

2:

2 2 1

8 1

9

3

3,2 , 3,2 , 3, 2 , 3, 2

x yy x

x y

y

x

x

x

x

SS

More Examples

Answer odd-numbered items in Leithold

Exercise 9.1, page 516-517

Exercise 9.3, page 534-536

Check your answers on A-63 and A-64.

End of Chapter 5.1