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Literal Equation Solver

Rearrange Formulas · Solve for Any Variable · Step-by-Step

Select any built-in formula (physics, geometry, finance, algebra) or enter a custom one. Choose the variable to isolate, see every algebraic step, then compute numerically.

Quick Examples

Formula Reference Library

All 20+ built-in formulas with their solvable variables.

Category	Formula	Description	Solve for

What Are Literal Equations?

A literal equation is any equation that contains two or more variables (letters representing quantities). Unlike standard equations with a single unknown, literal equations express universal relationships — the same formula holds true for every valid set of variable values. Examples are everywhere in science and mathematics: F = ma (Newton's second law), A = πr² (area of a circle), PV = nRT (ideal gas law), and I = Prt (simple interest).

The key skill with literal equations is rearranging them — isolating a specific variable on one side of the equation so it can be computed from the others. This process is also called solving for a variable, making a variable the subject, or transposing a formula.

Why Rearranging Formulas Matters

In physics, chemistry, engineering, and finance, you rarely measure every quantity directly. You measure the ones you can and calculate the rest from formulas. Consider Newton's second law, F = ma:

Know F and a, need m? Rearrange to m = F / a.
Know F and m, need a? Rearrange to a = F / m.

The same formula serves three different problems. Without rearrangement skills, every "variant" would need to be memorized separately — an impractical approach given thousands of scientific formulas.

Step-by-Step Method: Inverse Operations

The systematic approach to rearranging any formula is to apply inverse operations to both sides, working outward from the target variable:

Addition/Subtraction: To undo + b, subtract b from both sides.
Multiplication/Division: To undo × m, divide both sides by m.
Powers: To undo x², take the square root of both sides.
Roots: To undo √x, square both sides.
Logarithms: To undo log x, raise the base to both sides (exponentiate).

Worked Example: Solving d = ½at² for t

Starting formula: d = (1/2) a t²

Step 1: Multiply both sides by 2: 2d = a t²
Step 2: Divide both sides by a: 2d / a = t²
Step 3: Take square root: t = √(2d / a)

Since time must be non-negative, we take the positive root: t = √(2d / a).

Real-World Applications

Literal equations and formula rearrangement appear in virtually every quantitative field:

Physics: Kinematics (solving for time or initial velocity), optics (focal length), thermodynamics (ideal gas law).
Chemistry: Molarity, stoichiometry, ideal gas calculations.
Engineering: Stress/strain, Ohm's law, fluid dynamics (Bernoulli's equation).
Finance: Compound interest (solving for rate or time), loan amortization, present/future value.
Geometry: Solving for a dimension given area or volume.

Formula Reference: All Built-in Formulas

Formula	Name	Rearranged Forms
v = u + at	Velocity-time (kinematics)	u = v−at, a = (v−u)/t, t = (v−u)/a
d = ½at²	Distance (uniform acceleration)	a = 2d/t², t = √(2d/a)
F = ma	Newton's 2nd Law	m = F/a, a = F/m
E = mc²	Mass-energy equivalence	m = E/c²
KE = ½mv²	Kinetic energy	m = 2KE/v², v = √(2KE/m)
A = πr²	Circle area	r = √(A/π)
C = 2πr	Circle circumference	r = C/(2π)
A = ½bh	Triangle area	b = 2A/h, h = 2A/b
V = lwh	Rectangular volume	l = V/(wh), w = V/(lh), h = V/(lw)
V = (4/3)πr³	Sphere volume	r = (3V/(4π))^(1/3)
A = s²	Square area	s = √A
PV = nRT	Ideal Gas Law	P, V, n, T each solvable
I = Prt	Simple Interest	P = I/(rt), r = I/(Pt), t = I/(Pr)
A = P(1+r)ⁿ	Compound Interest	P = A/(1+r)ⁿ, r = (A/P)^(1/n)−1
y = mx + b	Slope-intercept	x = (y−b)/m, m = (y−b)/x, b = y−mx
m = (y₂−y₁)/(x₂−x₁)	Slope formula	y₁, y₂, x₁, x₂ each solvable

Frequently Asked Questions

What is a literal equation?

A literal equation is an equation with two or more variables representing physical or mathematical quantities. Examples include F = ma, A = πr², and PV = nRT. Solving a literal equation means isolating one specific variable on one side, expressing it in terms of all the others.

How do you rearrange a formula to solve for a variable?

Use inverse operations on both sides of the equation. If the variable is being added, subtract; if multiplied, divide; if squared, take the square root. Work step by step, applying each inverse operation to every term on both sides, until the target variable is alone on one side.

What is the difference between a literal equation and a regular equation?

A regular equation like 2x + 3 = 7 has one unknown and yields a single numeric answer. A literal equation has multiple variables; its solution is another formula showing one variable in terms of the rest. The result is universally valid for all values, not just a single number.

Why is rearranging formulas important in science and engineering?

Scientists and engineers always know some quantities and need to find others. A single formula like F = ma can solve for force, mass, or acceleration depending on which are known. Without rearrangement, every variant would have to be memorized separately — impractical given the vast number of formulas in physics, chemistry, and engineering.

Can this solver handle non-linear equations like A = πr²?

Yes. The solver handles common non-linear rearrangements: square roots (A = πr² → r = √(A/π)), cube roots (sphere volume for r), nth roots (compound interest for r), and squared terms (kinetic energy for v). Each step is displayed in full.

How does the numeric evaluator work?

After displaying the rearranged formula, the evaluator shows an input for each known variable. Enter their values and click Compute. The tool substitutes those values into the rearranged formula and returns the numerical result, bridging symbolic manipulation with practical calculation.

What formulas are included in the built-in library?

The library covers 20+ formulas: Physics (v = u + at, d = ½at², F = ma, E = mc², KE = ½mv²), Geometry (A = πr², A = ½bh, V = (4/3)πr³, C = 2πr, V = lwh, A = s²), Finance (A = P(1+r)ⁿ, I = Prt), Algebra (y = mx+b, slope formula), and Chemistry (PV = nRT).