5.1 systems of equations in two unknowns
TRANSCRIPT
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Chapter 5.1 Systems of Equations in
Two Unknowns
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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
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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
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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.
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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.
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Solving Systems of Equations
Methods for solving system of equations:
1. Substitution
2. Elimination
3. Using Matrices
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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End of Chapter 5.1