Algebra Know-It-ALL

314 Review Questions and Answers

Answer 12-9

If a and b are constants and x is the variable, then the standard form is

ax+b= 0

Any first-degree equation in one variable can always be put into this form. The variable might

be called something other than x, and the constants a and b might be expressed in terms of

other constants and numbers, but this basic form can always be derived.

Question 12-10

Put the equation derived in Answer 12-7, and shown in the last line of Table 20-1, into the

standard form for a first-degree equation in one variable.

Answer 12-10

Here’s the equation in the form we got when we solved for x in terms of the constants:

x= (b+ 3 c− 7)/(− 3 − 2 a)

We can multiply each side by (− 3 − 2 a) to obtain

(− 3 − 2 a)x=b+ 3 c− 7

Next, we can subtract the quantity (b+ 3 c− 7) from each side, getting

(− 3 − 2 a)x− (b+ 3 c− 7) = 0

Technically, this equation is in the standard form for a first-degree equation in one variable.

But we might want to rename the expressions made up of a,b,c, and numerals. Let’s use single

lettersp and q, as follows:

(− 3 − 2 a)=p

and

−(b+ 3 c− 7) =q

Now we have

px+q= 0

Chapter 13

Question 13-1

Figure 20-1 shows a mapping between sets. Four sets are identified, labeled A,B,C, and D.

The five points in set B represent all the elements in that set, and the five elements in set C

represent all the elements of that set. The dashed curves represent the entire mapping; they

“tell the whole story.” Furthermore, B⊂A and C⊂D. Which set is the maximal domain?

Which set is the co-domain? Which set is the essential domain? Which set is the range?