1001 Algebra Problems.PDF

926. 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, 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 equa-

tion in the system is obtained by multiplying both sides of the first equation by 3. Therefore, the two

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

927. 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..

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

Problem 911 yields the following solution:

So, the solution of the system is x= –1,y= 2.

928. 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..

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

lem 912 yields the following solution:

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

x

y

0 14

20

5

0 – 7

–

21

>> > >HHHH==^41

x

y

0

4

2

0

14

– 20

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

x

y

0

4

2

0

14

>>>HH= – 20 H

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

x

y

0

1

1

1

1

1

1

– 2

–– –

>> > >HHHH==

x

y 0

1

1

11

1

–

–

>> >HHH= –––^1

x

0 y

1

1

11

1

–

–

x >>>––HH H=

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

3

9

2

6

–

–

• 1

(^3) > H
9

2

6 0

–

> – H=

x

y

a

c

b

d

e

f

• 1

  
  

  
  

  
  

  HHH=

  
  

  
  

  
  

  
  

a

c

b

d

x

y

e

>>>HH H= f

ANSWERS & EXPLANATIONS–