1001 Algebra Problems.PDF

918. d.The solution to the matrix equation , where a, b, c, d, e,and fare real numbers, is

given by , provided that the inverse matrix on the right side exists. From Problem 886,

the given system can be written as the equivalent matrix equation. Note that since

det , it follows that does not exist, so we cannot apply the above principle. Rather, we

must inspect the system to determine whether there is no solution (which happens if the two lines are

parallel) or if there are infinitely many solutions (which happens if the two lines are identical). Multiply-

ing both sides of the second equation by 3 yields the equivalent equation 6x+ 3y= 9. Subtracting this

from the first equation yields the false statement 0 = –1. From this, we conclude that the two lines must

be parallel (which can also be checked by graphing them). Hence, the system has no solution.

919. a.The solution to the matrix equation , where a, b, c, d, e,and fare real numbers is

given by , provided that the inverse matrix on the right side exists. From Problem 887,

the given system can be written as the equivalent matrix equation. The solution is

therefore given by. Using the calculation for the inverse from Problem 903 yields

the following solution:

So, the solution of the system is x= – 171 ,y= – 252 .

920. a.The solution to the matrix equation , where a, b, c, d, e,and fare real numbers is

given by , provided that the inverse matrix on the right side exists. From Problem 888,

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

x

y

1

3

–

–

–

–

111

112

112

223

117

225

>> > >HHHH==

x

y

3

4

4

2

1

3

–

–

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

x

y

3

4

4

2

1

3

–

>>>HH H= –

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

6

2

3

1

• 1

  
  

  ^6231 H= 0 > H

  
  

  
  

x

y

6

2

3

1

8

>>>HH H= 3

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

ANSWERS & EXPLANATIONS–