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Reverse algebra is a mathematical derivation of traditional algebra, in which equations are manipulated from the inside out in order to solve for a particular variable or expression. This process often greatly reduces the number of written steps required to solve for a variable in an equation. While reverse algebra does not offer any new mathematical abilities, it can in certain situations improve efficiency.

## Concept

The essential concept behind reverse algebra is that, given a single equation which is to be solved for a single variable or expression, the algebraic steps needed to solve the equation can be performed in reverse, if the operational scope increment is also reversed. This allows for simple equations to be solved for a particular variable in a single written step, although some further steps may be needed to reduce it to simplest form. Because of the nature of this process of solving equations from the inside out, the resulting derived equation (solved for the variable desired) can begin to be produced while simultaneously being solved for. This overlap of tasks can dramatically improve efficiency, especially when solving equations by hand.

## Operational scope

The operational scope of an expression is the portion of the expression which will serve as an operand if an operation is applied. In traditional algebra, the operational scope of the expression on each side of an equation is always the entire expression. If an operation (such as multiplication) is performed, all of the expressions on both sides of the equation are operated on by the operand. After this operation is completed, the operational scope on each side is expanded to envelop the new expression. Because of this, operational scope is not an issue when using traditional algebra. However, with reverse algebra, the operational scope does not expand with each operation, and thus each applied operation is unaffected by previous operations.

## Procedure

The procedure of solving an equation for a variable or expression by hand using reverse algebra can be described as follows. In this description we will assume the variable for which the equation is to be solved is x. The expression which is the side of the equation containing x is isolated, and will be referred to here as the original expression. A new expression which will eventually be equal to the variable to be solved for is started as a blank expression, not equal to anything. Here this expression will be referred to as the new expression.

To actually produce a valid expression which is equal to x, operations are performed on x in its immediate scope in the original expression. For each operation performed on x in this expression, the same operation must be performed on the new expression. Because this expression is empty, each operation on it is therefore performed without an operand, and leave the operational scope for future operations unchanged (exclusive of the changes brought about by previous operations). This process is repeated until the original expression has been to reduced to simply x. At this point, the expression which is the opposite side of the equation containing the original expression is substituted in for the operational scope in the new expression. The new expression will then be equal to x.

Because operations on the scope in the solved equation do not affect the scope for future operations, each operation can be thought of as taking purely the scope itself as one of its operands. This allows for syntactic sugar in description, because the operational scope can be represented symbolically and described in equations, just as a variable can. Using this notation, the general rule for scope substitution as well as derivations of the rule can be expressed algebraically.

## Rules of scope substitution

Using ⊠ to represent the operational scope of the solved equation, and x to represent the variable to be solved for, the general rule for scope substitution is:

$f(x) = x \iff \boxtimes = f^{-1}(\boxtimes) \,$

From this fundamental rule can be derived more convenient rules for common arithmetical operations as follows.

\begin{align} x+b=x & \iff \boxtimes = -b+\boxtimes \\[10pt] bx=x & \iff \boxtimes = \frac\boxtimes b \\[10pt] \frac x b = x & \iff \boxtimes=b\boxtimes \\[10pt] x^n = x & \iff \boxtimes = \boxtimes^{1/n} \\[10pt] \frac{b}{cx} = x & \iff \boxtimes = \frac{b}{x\boxtimes} \end{align}

Note that in this list of rules, Latin letters such as b and c represent any combination of constants, variables or functions which do not include or depend on x. Each of these equivalencies implies that the variable x can, in the original equation, be replaced with its equivalence here if and only if the operational scope is replaced in the resultant equation with its equivalence. This process is repeated as needed, until the variable x is isolated on one side of the original equation. At this point, the other side of the original equation (which has not changed), is plugged in at the scope of the resultant equation. After this has been done, the equation has been completely solved for x, though it may need to be simplified.

## Example

Consider the following equation:

$a = x + 2 \,$

To solve this equation for x using reverse algebra, we first write the initial resultant equation in terms of the scope.

$x = \boxtimes \,$

We then take the original expression involving x in the original equation (x + 2), and apply the rules of equivalency until we isolate x. In this particular case, we would use the rule of addition. Thus we can replace (x + 2) with x in the original expression to obtain (x). But to do this we must also replace the ⊠ with (−2 + ⊠) in the resultant equation to obtain

$x = -2 + \boxtimes \,$

At this point, since we have isolated x in the original expression, we plug in the left side of the original equation (a) for the scope, to obtain

$x = -2 + a \,$

At this point the equation is solved.

 This article uses material from the Wikipedia article Reverse algebra, that was deleted or is being discussed for deletion, which is released under the Creative Commons Attribution-ShareAlike 3.0 Unported License.