Solving equations

For purposes of this course, we can consider an equation with one variable solved if the variable is alone on the left-hand side and does not appear on the right-hand side.

Here's a list of techniques for solving 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.

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 swap the direction of the inequality.

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This web page was written between 2007 and 2010 by Toby Bartels, last edited on 2010 November 28. Toby reserves no legal rights to it.

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