SAT — mathMedium6 min read

Literal Equations and Formulas

Learn to rearrange formulas and solve literal equations for the Digital SAT. Isolate any variable with step-by-step methods.

Tutor AI Team2026-02-25T01:07:10.179Z

A literal equation is an equation that contains two or more variables. Instead of solving for a numerical answer, you solve for one variable in terms of the others. The SAT frequently asks you to rearrange formulas — isolating a specific variable from a physics formula, geometry formula, or a general algebraic expression.

This skill is essential because many SAT problems present a formula and then ask you to express one variable in terms of others. Examples include rearranging $d = rt$ to find $t$ , or isolating $w$ from $P = 2l + 2w$ .

Core Concepts

What Is a Literal Equation?

A literal equation contains multiple variables (letters) rather than a single unknown. Examples:

$d = rt$ (distance = rate × time)
$A = \frac{1}{2}bh$ (area of a triangle)
$C = \frac{5}{9}(F - 32)$ (Celsius to Fahrenheit conversion)
$y = mx + b$ (slope-intercept form)

The Key Principle

Treat the variable you're solving for as the "unknown" and treat every other variable as if it were a number. Then use the same inverse operations you'd use for a regular equation.

Solving for a Variable: Step by Step

Identify which variable to isolate.
Use inverse operations to get that variable alone.
Other variables are treated as constants.

Example: Solve $d = rt$ for $t$ .

Divide both sides by $r$ :

$t = \frac{d}{r}$

Example: Solve $A = \frac{1}{2}bh$ for $h$ .

Multiply both sides by 2: $2A = bh$

Divide by $b$ : $h = \frac{2A}{b}$

Formulas with Multiple Steps

Example: Solve $P = 2l + 2w$ for $w$ .

Subtract $2l$ : $P - 2l = 2w$

Divide by 2: $w = \frac{P - 2l}{2}$

Formulas with the Variable in a Fraction

Example: Solve $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$ for $c$ .

$\frac{1}{c} = \frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}$

Take the reciprocal: $c = \frac{ab}{a + b}$

Formulas with the Variable Inside Brackets

Example: Solve $C = \frac{5}{9}(F - 32)$ for $F$ .

Multiply by $\frac{9}{5}$ : $\frac{9C}{5} = F - 32$

Add 32: $F = \frac{9C}{5} + 32$

Formulas with the Variable Squared

Example: Solve $A = \pi r^2$ for $r$ (where $r > 0$ ).

Divide by $\pi$ : $\frac{A}{\pi} = r^2$

Take the square root: $r = \sqrt{\frac{A}{\pi}}$

Strategy Tips

Tip 1: Treat Other Variables Like Numbers

If you're solving $2x + 3y = 12$ for $x$ , pretend $3y$ is just a number. Subtract it: $2x = 12 - 3y$ . Divide by 2: $x = \frac{12 - 3y}{2}$ .

Tip 2: Be Systematic

Follow the same steps as a regular equation: undo operations in reverse order.

Tip 3: Watch for the Variable in the Denominator

If the variable you want is in a denominator, multiply both sides by that variable to "free" it.

Tip 4: Factor When the Variable Appears More Than Once

If the target variable appears in multiple terms, collect those terms and factor it out.

Example: Solve $ax + bx = c$ for $x$ .

Factor: $x(a + b) = c$

Divide: $x = \frac{c}{a + b}$

Tip 5: Check Your Answer by Substituting

Plug your rearranged formula back into the original to verify it's consistent.

Worked Example: Example 1

Problem

The formula $V = lwh$ gives the volume of a rectangular prism. Solve for $h$ .

Divide both sides by $lw$ :

$h = \frac{V}{lw}$

Solution

Worked Example: Example 2

Problem

Solve $y = mx + b$ for $m$ .

Subtract $b$ : $y - b = mx$

Divide by $x$ : $m = \frac{y - b}{x}$

Solution

Worked Example: Example 3

Problem

Solve $S = \frac{n(a + l)}{2}$ for $a$ .

Multiply by 2: $2S = n(a + l)$

Divide by $n$ : $\frac{2S}{n} = a + l$

Subtract $l$ : $a = \frac{2S}{n} - l$

Solution

Worked Example: Example 4

Problem

Solve $T = 2\pi\sqrt{\frac{L}{g}}$ for $L$ .

Divide by $2\pi$ : $\frac{T}{2\pi} = \sqrt{\frac{L}{g}}$

Square both sides: $\frac{T^2}{4\pi^2} = \frac{L}{g}$

Multiply by $g$ : $L = \frac{gT^2}{4\pi^2}$

Solution

Worked Example: SAT-Style

Problem

The equation $3x - 2y = 18$ relates $x$ and $y$ . Which of the following expresses $y$ in terms of $x$ ?

Subtract $3x$ : $-2y = 18 - 3x$

Divide by $-2$ : $y = \frac{18 - 3x}{-2} = \frac{3x - 18}{2}$

Or equivalently: $y = \frac{3}{2}x - 9$

Solution

Practice Problems

Problem 1

Solve $E = mc^2$ for $m$ .

Problem 2

Solve $ax + by = c$ for $y$ .

Problem 3

Solve $\frac{x}{a} + \frac{y}{b} = 1$ for $y$ .

Problem 4

Solve $I = \frac{V}{R}$ for $R$ .

Problem 5

Solve $s = ut + \frac{1}{2}at^2$ for $a$ .

Problem 6

Solve $3(x + k) = 2(y - k)$ for $k$ .

Want to check your answers and get step-by-step solutions?

Common Mistakes

Not distributing before isolating. If the target variable is inside brackets, you may need to expand first.
Dividing incorrectly with multi-term expressions. In $w = \frac{P - 2l}{2}$ , you divide the entire expression $P - 2l$ by 2, not just $P$ .
Forgetting that square root introduces ±. When taking a square root, consider context. For physical quantities (lengths, areas), take the positive root.
Not factoring when the variable appears twice. If $x$ appears in two terms, you must factor it out.
Flipping only part of a fraction. When taking the reciprocal of $\frac{a + b}{c}$ , you get $\frac{c}{a + b}$ , not $\frac{c}{a} + \frac{c}{b}$ .

Frequently Asked Questions

How are literal equations different from regular equations?

The technique is identical. The only difference is that your answer contains variables instead of a number.

Will I need to rearrange complex physics formulas on the SAT?

The SAT uses formulas that look complex but typically require only two or three algebraic steps. You won't need advanced physics knowledge.

What if the variable I'm solving for appears more than once?

Collect all terms containing that variable on one side, factor it out, then divide.

Do I need to memorise any formulas for the SAT?

The SAT provides a reference sheet with key geometry formulas. However, you should know $y = mx + b$ , $d = rt$ , and basic area/volume formulas from memory.

Can I cross-multiply with literal equations?

Yes, if the equation has one fraction on each side, cross-multiplying works perfectly.

Key Takeaways

✓
Literal equations use the same inverse-operation method as regular equations.
✓
Treat all other variables as constants — focus only on the one you're solving for.
✓
When the target variable appears multiple times, collect its terms and factor.
✓
Clear fractions by multiplying by the denominator.
✓
Check your rearrangement by substituting back.
✓
This skill is highly testable on the SAT — practise with real formulas.

Tags:

#sat#math#algebra#formulas#literal-equations

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Core Concepts

What Is a Literal Equation?

The Key Principle

Solving for a Variable: Step by Step

Formulas with Multiple Steps

Formulas with the Variable in a Fraction

Formulas with the Variable Inside Brackets

Formulas with the Variable Squared

Strategy Tips

Tip 1: Treat Other Variables Like Numbers

Tip 2: Be Systematic

Tip 3: Watch for the Variable in the Denominator

Tip 4: Factor When the Variable Appears More Than Once

Tip 5: Check Your Answer by Substituting

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: Example 4

Worked Example: SAT-Style

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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