Algebra Know-It-ALL

196 First-Degree Equations in One Variable

The right-hand distributive law for division over addition, applied to the expression on the right side of

the equation, gives us

x=a/2+ (−b)/2+c/2+ (−12)/2

We can simplify the right-hand side by changing all the negative additions back to subtractions, and then

dividing out the numeral quotient. This gives us

x=a/2−b/2+c/2− 6

Products and Ratios

Let’s see what happens when the quantities on either side of an equation are multiplied by

constants, divided by nonzero constants, or both.

Examples

Here are five first-degree equations that contain a variable x multiplied and/or divided by

constants.

4 x= 0

x / 7 = 2

2 x/a=b

5 abx=c

3 x/(4a)= 3

Using the rules from Chap. 9, we can manipulate these equations to get x alone on the left

side, and the constants all by themselves on the right. That solves the equations. Here are the

results.

x= 0

x= 14

x=ab/2

x=c/(5ab)

x= 4 a

Are you confused?

If you can’t see straightaway how these solutions are derived, Tables 12-6 through 12-10 show how

the equations can be solved, step-by-step. Note that in the third and fifth original equations above

(and in Tables 12-8 and 12-10), we must not let a equal 0. Also, in the fourth solution equation (and

in Table 12-9), we must never allow either a or b to equal 0.