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 solutions are easy to identify, preferably a formula (an equation in which one 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 (at least whenever both sides of the equation are defined, which is always the case for linear equations), and a contradiction is an equation that's always false.

There is another method that can be used when the variable appears exactly once in the entire equation (so in particular the other side of the equation will be constant):

If the variable is on the right-hand side, then swap the sides.
Simplify the constant side (which is now the right-hand side).
Identify the last operation performed on the left-hand side and apply the reverse operation to both sides (and simplify them).
Repeat the previous step until the variable is alone on its side (which is the left-hand side).

So for example, if the last thing that is done on the left-hand side is to add 5, then you start solving by subtracting 5 from both sides.

You can also apply some tricks to make things go easier if you wish. For example, if the equation has fractions, then you may want to multiply both sides by a commmon denominator. Or if the coefficient on the variable is negative once the variable appears only on the left-hand side, then you may wish to subtract this variable term from both sides and ultimately get the variable on the right-hand side. (You can always swap sides at the end.) Ultimately, as long as you are using techniques from the list near the top of this page, if you manage to solve the equation with your chosen techniques, then you did it correctly.

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

If you multiply or divide both sides by a negative constant, 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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This web page was written between 2007 and 2018 by Toby Bartels, last edited on 2018 August 2. Toby reserves no legal rights to it.

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