19. Literal Equations, Formulas, and Units

Literal equations are ordinary equations with several letters. The requested letter is the variable; every other letter behaves like a known number.

Example 1 - simple interest:
\[ A=P(1+rt). \]
Solve for \(r\). Divide by \(P\):
\[ \frac{A}{P}=1+rt. \]
Subtract 1 and divide by \(t\):
\[ r=\frac{A/P-1}{t}=\frac{A-P}{Pt}. \]

Example 2 - variable in two terms:
\[ y=mx+b\quad\text{solve for }x. \]
Subtract \(b\), then divide by \(m\):
\[ x=\frac{y-b}{m}. \]

Example 3 - temperature:
\[ F=\frac95C+32. \]
Subtract 32 before multiplying by \(5/9\):
\[ C=\frac59(F-32). \]
The parentheses are essential; \(5F/9-32\) is a different expression.

Track units as if they were algebraic factors. Converting 55 miles per hour to miles per minute:
\[ 55\frac{\text{mi}}{\text{hr}}\cdot\frac{1\text{ hr}}{60\text{ min}} =\frac{11}{12}\frac{\text{mi}}{\text{min}}. \]
The hours cancel. If unwanted units do not cancel, a conversion factor is upside down.

When a formula contains a square, the context may select only the positive root. From \(A=\pi r^2\), a physical radius is \(r=\sqrt{A/\pi}\), not a negative length.

Common trap: moving a term across an equals sign by changing its sign without performing the same operation on the entire side.

Advanced mastery lab

Learning targets

• treat non-target letters as known quantities
• factor a target variable appearing in multiple terms
• use units and physical constraints to validate a formula

Visual model

How to read this SVG. Keep symbolic rearrangement ahead of number substitution. Units travel through the same operations and expose a reversed conversion, missing power, or dimensionally impossible result.

Decision protocol

1. Identify the target letter and operations attached to it.
2. Undo outer operations while preserving grouped quantities.
3. If the target appears twice, collect its terms and factor it.
4. Check dimensions, domain restrictions, and physical sign constraints.

Decision table

Timed transfer challenge: Isolate a geometric dimension

Target time. 4 minutes, including a dimensional check.

Problem. Solve \(V=\pi r^2h\) for \(h\). State the units if \(V\) is in cubic centimeters and \(r\) is in centimeters.

Solution.

1. The target \(h\) is multiplied by \(\pi r^2\).
2. Divide by the complete factor: \(h=\frac{V}{\pi r^2}\).
3. Units simplify as \(\text{cm}^3/\text{cm}^2=\text{cm}\).
4. The result has length units, as a height should.

Answer. \(h=\frac{V}{\pi r^2}\), in centimeters

Second independent check

Retrieval and reflection

Give a unit mismatch that would prove a formula rearrangement wrong.

Mastery standard. Complete the work accurately, explain the decisive step, and verify the answer using the original condition or an independent representation.