Overview:
Identity and equality properties in mathematics are some of the keys to solving algebraic equations. Those properties allow operations to be performed with numbers and variables. Some of them may have been learned in lower grades, without symbols to represent them.
What Are Some Identity Properties?
Zero in addition has a special property. Any number a +0 = a, and so does 0 +a equal a. For example, the equation b + 15 = 15 is solved by 0, because 0 +15 =15. The number 1 is the identity for multiplication, because any number multiplied by 1 is that number. Also, any number multiplied by zero will equal zero.
What Are Some Equality Properties?
The equality properties are always true for any numbers, a, b, or c. The reflexive property is that any number a is equal to itself. The symmetric property states that if a = b, then b = a. The transitive property states that if a = b and b=c, then a=c. The substitution property states that if a = b, then a may be replaced by b. Also, if a =b, then a +c = b + c, a-c = b-c, a∙c = b∙c, and as long as c does not equal 0, a/c = b/c.
How Are They Used?
Identity and equality properties can be used to solve many problems. Suppose the problem is to solve 6(12-48/4) +9∙1. First, 12-48/4 can be solved as 12-12 and then substituted so that the expression reads as 6 (12-12) + 9∙1. Now 6(0) can be substituted, and solved as 0, and 9∙1 can be solved as 9 because of the identity property for multiplication. Because 0 is the identity for addition, the entire expression can be solved as 9 +0 or 9.
Why Are the Properties Important?
The identity and equality properties are important tools to use when solving equations. Suppose an equation is 3 +x = 12, so 3 – 3 +x = 12 -3. Solving both sides of the equation, 3-3 equals 0, so 0 +x = 12-3. # is subtracted from both sides, using the equality property for subtraction, as well as the additive property for 0, and then x = 12-3 =9 is substitution.
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