Algebra Know-It-ALL

340 Review Questions and Answers

4 z= 13 − 2 x+y

4 × 1 = 13 − 2 × 6 + 3

4 = 13 − 12 + 3

4 = 13 + (−12)+ 3

4 = 4

Question 18-10

Suppose that when we checked our solutions in the preceding answer, we found that they did

not work out. What could we do then?

Answer 18-10

Whenever we check solutions and discover that they don’t work out, it means we’ve made a

mistake somewhere. In that case, we must go back through each step in our solution process

and find the mistake. We could also start all over and approach the problem in another way.

For example, we could eliminate a different variable in the beginning, and/or use different

equations to get the two-by-two linear system in the intermediate phase. After that, of course,

we’d have to check our solutions again for correctness.

Chapter 19

Question 19-1

Here is the three-by-three linear system taken from Answer 18-4:

4 x− 4 y− 4 z= 8

−x+ 2 y+ 5 z= 5

2 x−y+ 4 z= 13

How can we arrange these equations into a matrix that can be manipulated to solve this system?

Answer 19-1

We take the coefficients, remove the variables, and place the remaining numerals neatly in

the cells of a table. The first equation is represented by the top row of the matrix, the second

equation is represented by the middle row, and the third equation is represented by the bot-

tom row. The result looks like this:

(^4) − 4 − 4 8
− 1 255
2 − 1 413
Question 19-2
What general procedure can we use to solve the three-by-three system, starting with this
matrix?