Literal Equation
Pre-AlgebraA literal equation is an equation that contains two or more variables (letters), such as a formula that can be rearranged to solve for any one variable.
Definition
A literal equation is an equation with more than one variable (letter). Formulas are literal equations. You can rearrange them to solve for any variable you want.
Example
The perimeter formula $P = 2l + 2w$ has three variables. You can solve for $l$: subtract $2w$ from both sides and divide by $2$, giving $l = (P-2w)/2$.
Key Insight
Every formula you use in science and math is a literal equation. Rearranging formulas is a key skill for real-world problem solving.
Definition
A literal equation contains two or more variables. Solving a literal equation for a specific variable means isolating that variable using the same inverse-operation techniques as with numerical equations, treating all other variables as if they were constants.
Example
Solve $d = rt$ for $t$. Divide both sides by $r$: $t = d/r$. Solve $A = (1/2)bh$ for $h$: multiply both sides by $2$: $2A = bh$. Divide by $b$: $h = 2A/b$.
Key Insight
When rearranging a literal equation, the goal is not a number but an expression. The variable you solved for is now defined in terms of the others.
Definition
A literal equation is a polynomial (or rational) identity in multiple indeterminates, solved for one indeterminate in terms of the others. In the context of implicit differentiation, a two-variable equation $F(x, y) = 0$ can be solved for $dy/dx = -(dF/dx)/(dF/dy)$ by the implicit function theorem, provided $dF/dy \neq 0$.
Example
Rearranging the ideal gas law $PV = nRT$ for $T$: $T = PV/(nR)$. This expresses $T$ as a rational function of $P$, $V$, $n$, and $R$, with the restriction $nR \neq 0$.
Key Insight
The implicit function theorem generalizes the rearrangement of literal equations to nonlinear settings: near a point where $dF/dy \neq 0$, a curve $F(x, y) = 0$ locally defines $y$ as a smooth function of $x$.