Mathematical statements with one or more variables, or unknown numbers, are called open sentences. An open sentence is neither true nor false until the variable has been replaced by a value. Finding a replacement for the variable that results in a true sentence is called solving the open sentence. This replacement is called a solution of the open sentence.

Replace b in 6b + 7 = 37 with the valves 3, 4, 5, and 6. Then determine whether each replacement results in a true or false sentence.

Replace b with:

6b + 7 = 37

 True or False ?

3

6(3) + 7 =? 37 => 25 =/ 37

 False

4

6(4) + 7 =? 37 => 31 =/ 37

 False

5

6(5) + 7 =? 37 => 37 =  37

 True

6

6(6) + 7 =? 37 => 43 =/ 37

 False

Since b = 5 makes the sentence 6b + 7 = 37 true, the solution for 6b + 7 =37 is 5.

A set of numbers from which replacements for a variable may be chosen is called a replacement set.  The set of all replacements for the variable in an open sentence that results in a true sentence is called the solution set for the sentence.

 

A sentence that contains an equals sign, "=", is called an equation and sometimes may be solved by simply applying the order of operations. A sentence having the symbols < or > is called an inequality.

 

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