An equation is a mathematical statement that two things are equal to each other or that two things are conditionally equal to each other. If an equation is true for all values of any variables in the equation, it is called an identity. Identities are sometimes written using the symbol ≡ to indicate that it is always true.
|a + 0 ≡ a||The additive identity|
|a · 1 ≡ a||The multiplicative identity.|
|sin(θ)2 + cos(θ)2 ≡ 1||The Pythagorean Identity is one of the trigonometric identities.|
|Table 1: Examples of identities|
If an equation is not true for all values of the variables, it is called a conditional equation. The values of the variables for which the equation is true is called the solution of the equation.
|x = 5||This is a simple equation.||This is equation is true only when x = 5.|
|y = x + 1||This is a linear equation.||This equation is true for a set of ordered values (x,y) such that y = x + 1.|
|a2 + 2a + 4 = 0||This is a quadratic equation.||This equation is true only when a = -2|
An equation can be solved using the various property of real numbers and the properties of equality. When solving a one variable equation, the goal is to get the variable by itself on one side of the equation.
|x+5=2||This is the equation to solve.|
|(x+5)-5=2-5||Apply the Subtraction property of equality.|
|(x+5)+(-5))=2-5||Apply the Definition of subtraction.|
|x+(5+(-5))=2-5||Apply the Distributive property of multiplication over addition and subtraction.|
|x+0=-3||Simplify both sides of the equation.|
|x=-3||Apply the Property of addition by zero.|
|Table 1: Solving an equation.|
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