 Equations
 Solutions and Equivalent Equations
 Solving Equations
Inside Collection (Textbook): Fundamentals of Mathematics
Summary:
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to solve equations of the form
An equation is a statement that two algebraic expressions are equal.
The following are examples of equations:
Notice that
The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations. There are two additional types of equations. They are examined in courses in algebra, so we will not consider them now.
The set of values that, when substituted for the variables, make the equation true, are called the solutions of the equation.
An equation has been solved when all its solutions have been found.
Verify that 3 is a solution to
When
Verify that
When
Verify that 5 is not a solution to
When
Verify that 2 is a solution to
When
Verify that 5 is a solution to
Substitute 5 into
Verify that
Substitute 5 into
Verify that 0 is a solution to
Substitute 0 into
Verify that 3 is not a solution to
Substitute 3 into
Verify that 1 is a solution to
Substitute 1 into
Some equations have precisely the same collection of solutions. Such equations are called equivalent equations. For example,
We know that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side.
This number  is the same as  this number 
↓  ↓  ↓ 

=  4 

=  11 

=  1 
From this, we can suggest the addition/subtraction property of equality.
Given any equation,
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
Subtraction is used to undo an addition.
Addition is used to undo a subtraction.
The procedure is illustrated in the problems of (Reference).
Use the addition/subtraction property of equality to solve each equation.
4 is associated with
Check: When
The solution to
Check: When
becomes
a true statement.
The solution to
6 is associated with
This is equivalent to
Check: When
becomes
,
a true statement.
The solution to
Check: When
becomes
,
a true statement.
The solution to
Check: When
becomes
a true statement.
The solution to
Check: When
becomes
,
a true statement.
The solution to
For the following 10 problems, verify that each given value is a solution to the given equation.
Substitute
Substitute
Substitute
Substitute
Substitute
Solve each equation. Be sure to check each result.
((Reference)) Is
rate
((Reference)) Convert
((Reference)) 0.4% of what number is 0.014?
3.5
((Reference)) Use the clustering method to estimate the sum:
((Reference)) Combine like terms:
"Used as supplemental materials for developmental math courses."