Addition/subtraction property of equality
From this, we can suggest the
addition/subtraction property of equality .
Given any equation,
- We can obtain an equivalent equation by
adding the
same number to
both sides of the equation.
- We can obtain an equivalent equation by
subtracting the
same number from
both sides of the equation.
The idea behind equation solving
The idea behind
equation solving is to isolate the variable on one side of the equation. Signs of operation (+, -, ⋅,÷) are used to associate two numbers. For example, in the expression
, the numbers 5 and 3 are associated by addition. An association can be
undone by performing the opposite operation. The addition/subtraction property of equality can be used to undo an association that is made by addition or subtraction.
Subtraction is used to undo an addition.
Addition is used to undo a subtraction.
The procedure is illustrated in the problems of
[link] .
Sample set b
Use the addition/subtraction property of equality to solve each equation.
.
4 is associated with
by addition. Undo the association by
subtracting 4 from
both sides.
Check: When
,
becomes
The solution to
is
.
. 8 is associated with
by subtraction. Undo the association by
adding 8 to
both sides.
Check: When
,
becomes
a true statement.
The solution to
is
.
. Before we use the addition/subtraction property, we should simplify as much as possible.
6 is associated with
by addition. Undo the association by
subtracting 6 from
both sides.
This is equivalent to
.
Check: When
,
becomes
,
a true statement.
The solution to
is
.
. Begin by simplifying the left side of the equation.
1 is associated with
by addition. Undo the association by
subtracting 1 from
both sides.
Check: When
,
becomes
,
a true statement.
The solution to
is
.
. In this equation, the variable appears on both sides. We need to isolate it on one side. Although we can choose either side, it will be more convenient to choose the side with the larger coefficient. Since 8 is greater than 6, we’ll isolate
on the left side.
Since
represents
, subtract
from each side.
4 is associated with
by subtraction. Undo the association by
adding 4 to
both sides.
Check: When
,
becomes
a true statement.
The solution to
is
.
. -8 is associated with
by addition. Undo the by
subtracting -8 from
both sides. Subtracting -8 we get
. We actually
add 8 to both sides.
Check: When
becomes
,
a true statement.
The solution to
is
.
Practice set b
Exercises
For the following 10 problems, verify that each given value is a solution to the given equation.
,
Substitute
into the equation
.
is a solution.
,
,
Substitute
into the equation
.
is a solution.
,
,
Substitute
into the equation
.
is a solution.
,
,
Substitute
into the equation
.
is a solution.
,
,
Substitute
into the equation
.
is a solution.
,
Solve each equation. Be sure to check each result.
Calculator exercises
Exercises for review
(
[link] ) Is
an example of a ratio or a rate?
(
[link] ) Convert
% to a decimal.
(
[link] ) 0.4% of what number is 0.014?
(
[link] ) Use the clustering method to estimate the sum:
(
[link] ) Combine like terms:
.