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