Algebra Know-It-ALL

Part Two 313

Question 12-7

Suppose a,b, and c are constants, and x is a variable. How can we manipulate the follow-
ing equation so it contains x all by itself on the left side, and an expression containing
the constants without x on the right side, showing and justifying every step in an S/R
table?

x− 2 c+ 7 = 4 x+ 2 ax+b+c

Answer 12-7

We can morph the equation as described in Table 20-1. The end result is

x= (b+ 3 c− 7) / (− 3 − 2 a)

Question 12-8

We had better be careful about something in the last step of the process shown by Table 20-1.
What’s that?

Answer 12-8

We must require that 2a≠− 3. If we do not impose that restriction, we allow for the possibility
of division by 0 in the last step.

Question 12-9

What is the standard form for a first-degree equation in one variable?

Table 20-1. Equation morphing process for Answer 12-7. We solve for

x in terms of the constants. But there’s a catch, which is addressed in

Question and Answer 12-8.

Statements Reasons

x− 2 c+ 7 = 4 x+ 2 ax+b+c This is the equation we are given

x+ 7 = 4 x+ 2 ax+b+ 3 c Add 2c to each side

x= 4 x+ 2 ax+b+ 3 c− 7 Subtract 7 from each side

x− 4 x= 2 ax+b+ 3 c− 7 Subtract 4x from each side

x− 4 x− 2 ax=b+ 3 c− 7 Subtract 2ax from each side

x+ (− 4 x)+ (− 2 a)x=b+ 3 c− 7 Change subtractions to negative additions

on left side of equation

[1+ (−4)+ (− 2 a)]x=b+ 3 c− 7 Right-hand distributive law on left side of

equation

(− 3 − 2 a)x=b+ 3 c− 7 Simplify left side of equation

x= (b+ 3 c− 7)/(− 3 − 2 a) Divide each side by (− 3 − 2 a)