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:

Simplify either side (or both);
Add or subtract the same expression to both sides (and then simplify them);
Multiply or divide both sides by the same nonzero constant (and then simplify them);
Swap the sides.

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:

Simplify both sides (if necessary).
If there is a variable term on the right-hand side, then subtract this term from both sides (and simplify them).
If there is a constant term on the left-hand side, then subtract this term from both sides (and simplify them).
If there is now a coefficient on the variable on the left-hand side, then divide both sides by that coefficient (and simplify them).

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:

If you multiply or divide both sides by a negative number, then you must switch the direction of the inequality.

You also need to switch the direction if you swap the sides, although this point is much easier to remember.

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