An **equation** consists of two algebraic
expressions and the symbol
between them. The algebraic expressions contain variables and
constants. If the equation is true for all values of the variables
it is called an **identity.** An example
of an identity is

The focus on this page, however, is on equations that are true only
for some values of the variables. There may be several such
variables, but to begin with we assume there is only one, and we
usually call it . Figuring out for which values of an
equation is true is called **solving** the equation. A value of
that makes the equation true is called a **solution** of the equation. For example, the
equation

The fundamental principle of equation solving is based on the fact that after applying the same operation on both sides of the equation the solutions of the original equation are also solutions of the new equation. Think of the equation as one of those old fashioned scales that balance an initially unknown weight with a combination of known weights. If everything is in balance and you do whatever you do on both sides everything will still be in balance.

Simply put, the fundamental principle of equations solving is

** To solve an equation figure out what bothers you the most at the
moment and get rid of it by applying a suitable operation on both
sides of the equation.**

For example, consider the above mentioned equation We want to solve it. This means we want to obtain another equation of the form where the right hand side of the equation does not involve . Well, is not by itself in the original equation. It is multiplied with 3 and 4 has been added. Both the 3 and the 4 bother us. We could get rid of them in either sequence, but it's simpler to get rid of 4 first, by subtracting 4 on both sides of the equation. Since and this gives the new equation.

In the literature, like in your texbook, the simplicity and power of the fundamental principle is obscured by the fact that there is a long list of special cases. For example, just for linear equations our textbook lists: solving linear equations in standard form, linear equations in non-standard from, linear equations involving fractions, linear equations involving decimals, and linear equations--special cases. The issue is further confused by giving various names to "applying the same operation on both sides". For example, in our textbook there is the "addition property of equality" (meaning you can add the same term on both sides), the "multiplication property of equality" (meaning you can multiply with the same non-zero factor on both sides), etc.

This is like having a city guide that contains sections on how to walk along 13th East, how to walk along 7th East, how to walk across First South, and so on. Actually, all you need to know is how to walk. Similarly, to solve equations, all you need to understand and appreciate is the above mentioned principle. Once you do that solving equations is just a matter of practicing and gaining experience.

There is a subtlety that is hard to appreciate at first. "Doing the same thing on both sides" may introduce additional solutions (called spurious in this context). For example, consider the equation

This gives rise to an auxiliary principle of equation solving:

** After solving an equation check your solutions by
substituting them in the original equation.**

Checking your answers carefully not only eliminates spurious solutions but it also helps guard against errors.