Solving equations

A solution of an equation is an assignment of values to an equation's variables that make the equation come out true. Two equations are equivalent if they have the same solutions; every assignment of values to variables that makes either equation true must also make the other equation true. To solve an equation is to turn it into an equivalent equation whose the solutions are easy to identify, preferably a formula (where the variable is alone on the left-hand side and does not appear on the right-hand side).

Here's a list of techniques useful for solving equations by turning them into equivalent equations:

There are always more techniques, some of which are still being discovered.

A linear equation is an equation whose sides are both linear expressions. Linear equations in one variable can always be solved using this method:

At this point, you should have the answer, with the variable equal to a constant. Failing that, you might have a constant statement (with no variable in it), which will be either true or false; then that (‘True’ or ‘False’) is your answer. (Among equations, an identity is an equation that's always true, and a contradiction is an equation that's always false.)

Inequalities are solved in the same way as equations, except for this important point:

You also need to switch the direction if you swap the sides, although this point is much easier to remember.
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This web page was written between 2007 and 2014 by Toby Bartels, last edited on 2014 May 19. Toby reserves no legal rights to it.

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