Algebra Know-It-ALL

202 First-Degree Equations in One Variable

where k is a constant. Can this be rearranged into standard first-degree equation form? If so, how? If not,

why not? What is the significance of the result?

Solution

Because k is a constant, the quantity (8/k) must also be a constant, and can be treated as one. However,

k cannot be allowed to equal 0, because in the equation as stated, it appears in the denominator of a fraction.

With that in mind, you can add the quantity (8/k) to each side of the equation, getting

4/x− 8/k+ 8/k= 8/k

which simplifies to

4/x= 8/k

Using the rule of cross-multiplication, you know that

4 k= 8 x

Now you can add − 8 x to each side, getting

− 8 x+ 4 k=− 8 x+ 8 x

which simplifies to

− 8 x+ 4 k= 0

This is in standard first-degree form, ax+b= 0, if you let a=−8 and b= 4 k. Therefore, the original equa-

tion is a first-degree equation. At first glance, it might seem that the original equation can’t be first-degree,

because it looks as if x, which is the denominator of a fraction, is raised to the −1 power. But if a single-

variable equation can be converted into standard first-degree form, then it is a first-degree equation.

Here’s another challenge!

Suppose you come across this equation in a physics or engineering problem. Note the similarity of this to

the equation in the previous challenge:

4/x= 0

Can this be rearranged into standard first-degree equation form? If so, how? If not, why not? What is the

significance of the result?

Solution

You can rewrite this as

4/x= 0/1

Then you can try using the rule of cross-multiplication, getting

4 = 0 x