Difference between revisions of "Equation"

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Revision as of 18:19, 4 June 2015

An equation is a relation which states that two expressions are equal, identical, or otherwise the same. Equations are easily identifiable because they are composed of two expressions with an equals sign ('=') between them.

Equations are similar to congruences (which relate geometric figures instead of numbers) and other relationships which fall into the category of equivalence relations.

A unique aspect to equations is the ability to modify an original equation by performing operations (such as addition, subtraction, multiplication, division, and powers).

It's important to note the distinction between an equation and an identity. An identity in terms of some variables states that two expressions are equal for every value of those variables: for example,

$x^2 - y^2 = (x - y)(x + y)$

is an identity that is true regardless of the values of $x$ and $y$ (and indeed holds in a commutative ring). However,

$x^2 = 4$

is an equation that is true for some particular values of $x$.

In other words, one can say that an identity is a tautological equation

Linear equations

A linear equation is the simplest form of equations (with one or more variables). It has the form $ax + by + cz + ... = n$, where $a, b, c, ..., n$ are numbers and $x, y, z, ...$ are the variables. Some examples of linear equations are the following:

$2x + 5 = 20$

$2x + 4b = 16$

$ax + by = cz$

How to solve a linear equation with one variable

Linear equations with one variable are of the form $ax = b$, where $a, b$ are numbers and $x$ is a variable. The common recipe for solving linear equations with one variable is the following:

Solve the equation $5x + 4 = 24$

1. $5x + 4 = 24$ (We have to distinguish the terms. On the right side we put the variable terms and on the left side we put the numbers.)

2. $5x + 4 - 4 = 24 - 4$ (In order to achieve it, we have to add or subtract in both hand sides.)

3. $5x = 20$ (Now we have to convert the equation into the form $ax = b$. We shall proceed to divide both sides by $a=5$.)

4. $\dfrac {5x}{5} = \dfrac {20}{5}$

5. $x = 4$

Hence the solution to our equation is $\boxed{x = 4}$.

See also

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