Here are the definitions for the terms you've requested:
Solution Set
A solution set is a collection of all possible values that satisfy a given mathematical equation or inequality. For example, if you have the equation \( x + 2 = 5 \), the solution set would contain the value {3}, as that is the only value for \( x \) that satisfies the equation. In cases where an equation has multiple solutions, the solution set could contain more than one element. The solution set can be represented in various forms, including as a set of numbers, an interval, or even graphically on a number line.
Addition Property of Equality
The addition property of equality states that if you have two equal quantities, adding the same value to both sides of the equation does not change the equality. In symbolic terms, if \( a = b \), then \( a + c = b + c \) for any number \( c \). This property is fundamental in solving equations, as it allows us to isolate variables through addition.
Division Property of Equality
The division property of equality indicates that if two quantities are equal, dividing both sides of the equation by the same non-zero number will maintain the equality. In symbolic terms, if \( a = b \) and \( c \neq 0 \), then \( \frac{a}{c} = \frac{b}{c} \). This property is useful for simplifying equations and isolating variables during the solving process.
Identity
In mathematics, an identity is an equation that is true for all values of the variable involved. An example of an identity is the equation \( x + 0 = x \), which holds true for any value of \( x \). In the context of arithmetic operations, an identity element refers to a number that, when used in a specific operation with another number, leaves the other number unchanged. For example, the identity element for addition is 0, and for multiplication, it is 1.
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