C L Oates
8 May 2010
OCCC

Some Example Problems for Simultaneous Linear Equations

p. 506, #41

Abstract: $5/hr before 11 p.m., $7.50/hr after 11 p.m.
paid $30 for 5 total hours work.

Variables: Let x be the number of hours worked before 11 p.m..

Let y be the number of hours work after 11 p.m.

Equations: Since there were 5 total hours worked before and after 11 p.m.,

x + y = 5

The total pay of $30 can be expressed as

$5 x + $7.5 y = $30 .

The equations can be solved graphically, by substitution, or by elimination.

Solution by TI-83/84 calculator is also possible. These two equations can be expressed as an augmented 2 x 3 matrix as follows.

1 1 | 5
5 7.5 | 30

Solving this system of equations as described in the class handout (but with a 2 x 3 matrix, of course) yields

1 0 | 3
0 1 | 2 .

Solution: Therefore x = 3 and y = 2, but that is not the required answer. The required answer is, Three hours are worked before 11 p.m., and two hours are worked after 11 p.m.

 

p. 495, #17

Abstract: 315 million cats, dogs, and fish owned as pets; number of fish is 16 million less than total of dogs and cats; 17 million more cats than dogs.

Variables: Let f be the number of fish owned as pets.
Let c be the number of cats owned as pets.
Let d be the number of dogs owned as pets.

Equations: Since there were 315 million total cats, dogs, and fish,

f + c + d = 314 (in millions) .

c + d - 16 = f (again in millions)

c = d + 17

These equations can be re-written with the variables and their coefficients on the left and the constants in millions on the right as follows.

f + c + d = 314

-f + c + d = 16

+ c - d = 17

Now, re-writing the above equation as an augmented matrix, we have

1 1 1 | 314
-1 1 1 | 16
0 1 -1 | 17

Solving this system of equations as described in the class handout (this time with a 3 x 4 matrix) yields

1 0 0 | 149
0 1 0 | 91
0 0 1 | 74

and , therefore, f = 149, c = 91, and d = 74, all in millions.

The answer to the question, then is 149 million fish, 91 million cats, and 74 million dogs.

 

I hope these examples are helpful. Let me know if you have questions or find errors in the above.

Prof. Oates