#
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