4.3 cramer’s rule
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Transcript of 4.3 cramer’s rule
09.16.10/09.17.10 ACT OPENER
1. Multiply -3[4 -7 -½]
A. [-12 21 -1.5]B. [-12 21 1.5] C. [1 -10 -3.5]D. [10.5] E.
3. Solve the system of linear equations by graphing. x + 2y = -4 4y = 3x + 12
2. [2 3 4] + [-2 -3 -4] = ?
F. [-4 -6 -8]G. [-4 -9 -16]H. [0 0 0]J. [0]K.
5.1
32
12
4
3
2
x + 2y = -4
4y = 3x + 12
Announcements Test Corrections
You may correct 1 test per semester You will receive ½ credit Test 1 Corrections are Due: Wednesday,
September 22
Late Work You may turn in late assignments for ½ credit No work will be accepted 1 week past the due
date All late work from 8/11 – 9/16 is due
Wednesday, September 22
Remember to save your tests for the final exam to get one bonus point!
Cramer’s Rule (Section 4.3)
Gabriel Cramer was a Swiss mathematician (1704-1752)
Coefficient Matrices You can use determinants to solve a
system of linear equations. You use the coefficient matrix of the
linear system. Linear System Coeff Matrix
ax+by=ecx+dy=f
dc
ba
Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear System Coeff Matrix ax+by=e cx+dy=f
Find the second-order determinant.
If detA ≠ 0, then the system has exactly one solution.
dc
ba
bcaddc
baA det
Determinants
58
32det
458
732
yx
yxExample 1
bcaddc
baA det
)8)(3()5(2
2410
34
Cramer’s Rule for 2x2 System If detA ≠ 0, then the system has exactly one solution:
bcaddc
baAD det
Denominator
df
benx fc
eany
Linear Systemax+by=ecx+dy=f
Numerator of x Numerator or y
Solution of Linear System:
d
n
d
n yx ,
Cramer’s Rule for 2x2 System
Linear System ax+by=ecx+dy=f
A
df
be
d
nx x
det
A
fc
ea
d
ny y
det
constants y constantsx
Example 2 Solve the system: 8x+5y=2 2x-4y=-10
42
58
dc
baThe coefficient matrix is:
42)10()32()2)(5()4)(8(42
58det
A
And:
410
52
df
benx
and102
28
fc
eany
Solution: (-1,2)
410
52
xn
102
28
yn
)10)(5()4)(2( 50842
)2)(2()10)(8( 480
84
142
42
d
nx x
242
84
d
ny y
Example 2
Example 3 Solve the system: 3x + 7y = 11 8x + 5y = 13
58
73
dc
baThe coefficient matrix is:
415615)8)(7()5)(3(58
73det A
And:
513
711
df
benx
and138
113
fc
eany
Solution:
513
711xn
138
113yn
)13)(7()5)(11( 9155 36
)8)(11()13)(3( 8839 49
41
36
41
36
d
nx x
41
49
41
49
d
ny y
Example 3
41
49,
41
36
Example 4 Solve the system: 3x +4y = 2 5x – 7y = 17
75
43
dc
ba
The coefficient matrix is:
412021)5)(4()7)(3(75
43det
A
And:
717
42
df
benx
and
3 2
5 17y
a en
c f
Solution:
2 4
17 7xn
3 2
5 17yn
(2)( 7) (4)(17) 14 68 82
(3)(17) (2)(5) 51 10 41
822
41xnxd
41
141
ynyd
Example 4
2, 1
Exit Slip
Solve the system: 8x + 3y = 41 6x + 5y = 39