# Section 1-4

# Solving Equations with Variables on Both Sides

Sometimes, you get a linear equation in one variable, but with terms containing
the variable occuring on both sides of the equal sign. In this case, you have
to combine like terms first,
and then proceed to solve the equation using the addition, subtraction, multiplication
and division properties
of equality.

**Example 1: **

Solve for *p*:

** 21 – 3***p* = 4*p*

Start off by adding 3*p* to both sides.

**21 – 3***p* + 3*p* = 4*p* + 3*p*

**21 = 7***p*

(We are using the Distributive
Property here to add 4*p* + 3*p* and get 7*p*.)

Now we can divide through by 7.

**
***p* = 3

**Example 2: **

Solve for *r*:

** 3***r* + 4 + 6*r* = –23

Use the Commutative
and Associative Properties to group like terms, and then use the Distributive
Property to combine them.

**
(3***r* + 6*r*) + 4 = –23

**
9***r* + 4* = *–23

Next, subtract 4 from both sides.

**
9***r* + 4 – 4* = *–23 – 4

**
9***r = *–27

Finally, divide both sides by 9.

**
***r* = –3