1001 Algebra Problems.PDF

916. 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. From Problem 884,

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 900 yields

the following solution:

So, the solution of the system is x= –7,y= 5.

917. 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 885,

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 this 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). The

second equation in the system is obtained by multiplying both sides of the first equation by –2. The two

lines are identical, so the system has infinitely many solutions.

1

2

2

4

–

–

• 1

  
  

  H

  
  

  
  

1

2

2

4

– 0

–

> H=

x

y

1

2

2

4

3

6

–

>>>––HH H=

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

x

y

1

1

3

2

1

2

7

5

–

––

–

>> > >HHHH==

x

y

2

1

3

1

1

– 2

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

x

y

2

1

3

1

1

>>>HH H= – 2

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

ANSWERS & EXPLANATIONS–