Algebra Know-It-ALL

Standard form

Whenever you encounter an equation that can be morphed into the following form, then that
equation is a first-degree equation:

ax+b= 0

where x is the variable, and a and b are constants. This is called the standard form for a first-
degree equation in one variable. It’s possible that b can equal 0. If a= 0, then x disappears, the
statement ends up trivial or false, and it’s not a first-degree equation in one variable—because
there is no variable! Here are several examples:

x= 0

3 x= 0

− 4 x= 0

x+ 3 = 0

x− 4 = 0

5 x+ 2 = 0

5 x− 2 = 0

− 5 x+ 2 = 0

− 5 x− 2 = 0

Here’s a challenge!

Imagine that you are working on a problem in physics, engineering, or some other branch of science and
you come across this equation:

4/x− 8/k= 0

Combinations of Operations 201

Table 12-17. Streamlined process for solving the equation

6 a− 3 x/(bc)=− 24 d, provided b≠ 0 and c≠ 0.

Statements Reasons

6 a− 3 x/(bc)=− 24 d This is the equation we are given

− 3 x/(bc)=− 6 a− 24 d Subtract 6a from each side

3 x/(bc)= 6 a+ 24 d Multiply through by − 1

3 x= (6a+ 24 d)(bc) Multiply through by (bc)

3 x= 6 abc+ 24 dbc Right-hand distributive law for multiplication

over addition

3 x= 6 abc+ 24 bcd Commutative law for multiplication in second

addend on right side

x= (6abc+ 24 bcd)/3 Divide through by 3

x= 2 abc+ 8 bcd Right-hand distributive law for division over

addition