1001 Algebra Problems.PDF

923. c.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

891, the given system can be written as the equivalent matrix equation. The solu-

tion is, therefore, given by. Using the calculation for the inverse from Problem 907

yields the following solution:

So, the solution of the system is x= 2,y= –4.

924. b.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..

The solution is, therefore, given by. Using the calculation for the inverse from

Problem 908 yields the following solution:

So, the solution of the system is x= 2,y= 3.

925. 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. Note that since

det , it follows that does not exist. Therefore, we cannot apply the 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). Subtracting the

second equation from the first equation yields the false statement 0 = –3. 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.

3

3

2

2

• 1

  
  

  ^3322 H= 0 > H

  
  

  
  

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

x

y

1

2

0

1

2

1

2

3

–

––

>> > >HHHH==–

x

y

1

2

0

1

2

1

–

–

>> >HHH= –^1 –

x

y

1

2

0

1

2

1

–

–

= –

>>>HH H

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

x

y 10

4

0

2

4

–––

–

>> > >HHHH==^2121

x

y

0

2

1

1

4

–– 0

= –^1 –

>> >HHH

x

y

0

2

1

1

4

–– 0

= –

>>>HH H

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

ANSWERS & EXPLANATIONS–