Literal Equations on the SAT
Solve for the Target Variable
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Literal Equations
Literal equations show up any time a formula needs to be rearranged, which is common in science and real world contexts. You might solve the area formula for $r$, or rewrite a physics equation for time instead of distance.
The SAT uses these problems to test your algebra control. This lesson shows how to treat letters as numbers, how to factor the target variable cleanly, and how to check that your final expression makes sense.
A Simple Definition Unlocks Literal Equations
A literal equation is solved the same way as a normal equation, you isolate the variable you want. The difference is that the other letters stay in the answer, so you need to be careful with each algebra move.
The most common mistake is canceling terms that are not factors. When the target variable appears in more than one term, you usually need to factor it out before you can divide and isolate it.
Work Through Literal Equations Step by Step
Solve $A = \pi r^2$ for $r$, then find $r$ when $A = 50$.
Here is a classic SAT example using a formula you already know.
Start with the area formula
Divide both sides by $\pi$
Take the square root to undo the square.
Use Desmos to Check Literal Equations
Solve $A = \pi r^2$ for $r$, then find $r$ when $A = 50$.
Desmos is not ideal for symbolic rearrangement, but it can solve for a numeric value quickly if the equation includes numbers. This is useful when the SAT gives a numeric formula and asks for a specific value.
y = pi x^2
y = 50
Desmos is faster when the arithmetic is messy. Algebra is faster when the question asks for a symbolic rearrangement like solving for $r$ in terms of $A$.
Expert move: Graph the left and right sides directly (no need to isolate $y$), then click the intersection(s) and read the $x$-value(s). If Desmos gives decimals, convert to fractions when the choices are exact and keep only values that fit the domain or context.
When to skip Desmos: If the algebra is one or two steps, solve by hand and use Desmos only to verify; Desmos is best for messy coefficients or checking setup.
- Desmos features used: graphing, intersection point.
- Common mistake: using Desmos for a symbolic result instead of a numeric one.
Practice Literal Equations with SAT-Style Questions
Rearrange each formula to isolate the requested variable.
Solve for in terms of : .
Solve for in terms of : .
Solve for : .
Solve for in terms of and : .
Key Takeaways to Remember for Literal Equations
- Treat every other letter like a constant and isolate the target variable.
- Clear fractions and factor the target variable when needed.
- Desmos helps with numeric values, but algebra wins for symbolic rearrangement.
About the Instructor
