Algebra Know-It-ALL

198 First-Degree Equations in One Variable

Here’s a challenge!

Manipulate the following equation so it contains x all by itself on the left side, and an expression containing

the constants without x on the right side. Indicate, if applicable, which constants cannot equal 0.

3 abx/(4cd)=k^2

Solution

We must have c≠ 0 and d≠ 0 because, if either of them are allowed to equal 0, the left-hand side of the

equation becomes undefined. Let’s multiply the equation through by the quantity (4cd). We get

[3abx/(4cd)](4cd)= (4cd)k^2

which simplifies to

3 abx= 4 cdk^2

Now we can divide the entire equation through by the quantity (3ab). When we do this, we must insist

thata≠ 0 and b≠ 0. That produces

3 abx/(3ab)= 4 cdk^2 /(3ab)

which simplifies to

x= 4 cdk^2 /(3ab)

That does it! We don’t have to worry about the fact that one of the constants is squared. The square of

a constant is always another constant. The variable, x, is never raised to any power (other than the first

power), so the equation is a first-degree equation.

Combinations of Operations

In a first-degree equation that involves a single variable, constants can be added to or sub-

tracted from that variable, and the variable can also be multiplied or divided by nonzero

constants.

Examples

Here are some first-degree equations that involve combinations of sums, differences, products,

and ratios:

8 x− 4 = 0

18 x+ 7 =− 2

a− 3 x= 0

a− 5 + 15 x= 0