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SOLVING SYSTEMS OF LINEAR EQUATIONS
• An equation is said to be linear if every variable has degree equal to one (or zero)
• is a linear equation
• is NOT a linear equation
8254 zyx
832 13 zyx
Review these familiar techniques for solving 2 equations in 2 variables. The same techniques will be
extended to accommodate larger systems.
13
52
yx
yx
13
1536
yx
yx
13
2
yx
x
Times 3
Add
Substitute to solve: y=1
147 x
L1 represents line one L2 represents line two
13
52
yx
yx13
1536
yx
yx
13
2
yx
x
L1 is replaced by 3L1
L1 is replaced by L1 + L2
L1 is replaced by (1/7)L1
13
147
yx
x
13
52
yx
yx
1 3
14 7
y x
x
13
1536
yx
yx
13
2
yx
x
These systems are said to be EQUIVALENT because they have the SAME SOLUTION.
PERFORM ANY OF THESE OPERATIONS ON A SYSTEM OF LINEAR EQUATIONS TO
PRODUCE AN EQUIVALENT SYSTEM:
• INTERCHANGE two equations (or lines)
• REPLACE Ln with k Ln , k is NOT ZERO
• REPLACE Ln with Ln + cLm
• note: Ln is always part of what replaces it.
EXAMPLES:
is equivalent to
L1
L2
L3
L1
L3
L2
L1
L2
L3
L1
4L2
L3
is equivalent to
is equivalent to
L1
L2
L3 + 2L1
L1
L2
L3
253
343
0842
zyx
zyx
zyx
1 z
343 zyx
042 zyxReplace L1 with (1/2) L1
Replace L3 with L3 + L2
1
343
042
z
zyx
zyx
Replace L2 with L2 + L1
1z
3y
042 zyx
1
3
042
z
y
zyx
1
3
z
y
6x +4zReplace L1 withL1 + 2 L2
1
3
64
z
y
zxReplace L1 with L1 + - 4 L3
1
3
z
y
2x
253
343
0842
zyx
zyx
zyx
is EQUIVALENT to
1121
0111
2233
zyx
zyx
zyx
To solve the following system, we look for an equivalent systemwhose solution is more obvious. In the process, we manipulateonly the numerical coefficients, and it is not necessary to rewritevariable symbols and equal signs:
1121
0111
2233
zyx
zyx
zyx
This rectangular arrangement of numbers is called a MATRIX
1121
0111
2233
1121
0111
2011
2
Replace L1 with L1 + 2 L2
1121
0111
2233
1010
0111
2011
1
Replace L3 with L3 + L2
1121
0111
2233
1121
0111
2011
1010
2100
2011
1
Replace L2 with L2 + 1L1
1121
0111
2233
1121
0111
2011
1010
0111
2011
1010
2100
2011
1121
0111
2233
1121
0111
2011
1010
0111
2011
1121
0111
2233
1121
0111
2011
1010
2100
2011
1010
0111
2011
1121
0111
2233
1121
0111
2011
1010
2100
2011
1010
0111
2011
2100
1010
2011
Interchange L2 and L3
1121
0111
2233
1121
0111
2011
1010
2100
2011
1010
0111
2011
2100
1010
2011
Replace L1 with L1 + -1 L2
Replace L3 with -1 L3
Replace L2 with -1 L2
1121
0111
2233
1121
0111
2011
1010
2100
2011
1010
0111
2011
2100
1010
2011
1001
2100
1010
1121
0111
2233
1001
2100
1010
The original matrix representsa system that is equivalent to this final matrix whose solution is obvious
1121
0111
2233
1001 x
2100 z
1010 y
The original matrix representsa system that is equivalent to this final matrix whose solution is obvious
The zeros
The diagonal of ones
1001 x
2100 z
1010 y
Note the format of the matrix that yields this obvious solution:
2100
1010
1001
Whenever possible, aim for this format.