Equivalent equationsare related equations that have the same solution set. To
solve an equation, we usually start with the given equation and replace it with a series
of simpler equivalent equations. For example,
and Equivalent equations
are all equivalent, since each has the solution set
OBJECTIVE 3 Solve linear equations by using the addition and multiplica-
tion properties of equality. We use two important properties of equality to produce
equivalent equations.
536.
5 x+ 2 = 17, 5 x=15, x= 3
SECTION 2.1 Linear Equations in One Variable 49
Addition and Multiplication Properties of Equality
Addition Property of Equality
For all real numbers A, B, and C, the equations
and are equivalent.
That is, the same number may be added to each side of an equation without
changing the solution set.
Multiplication Property of Equality
For all real numbers Aand B, and for the equations
and are equivalent.
That is, each side of an equation may be multiplied by the same nonzero
number without changing the solution set.
AB ACBC
CZ0,
AB ACBC
Because subtraction and division are defined in terms of addition and multiplica-
tion, respectively, the preceding properties can be extended.
The same number may be subtracted from each side of an equation, and
each side of an equation may be divided by the same nonzero number, with-
out changing the solution set.
Using the Properties of Equality to Solve a Linear Equation
Solve
The goal is to isolate xon one side of the equation.
Combine like terms.
Add 5 to each side.
Combine like terms.
Subtract 6xfrom each side.
Combine like terms.
Divide each side by
Check by substituting - 3 for xin the originalequation.
x=- 3
- 4.
- 4 x
- 4
=
12
- 4
- 4 x= 12
2 x- 6 x= 12 + 6 x- 6 x
2 x= 12 + 6 x
2 x- 5 + 5 = 7 + 6 x+ 5
2 x- 5 = 7 + 6 x
4 x- 2 x- 5 = 4 + 6 x+ 3
4 x- 2 x- 5 = 4 + 6 x+ 3.
EXAMPLE 2
