Solving for a Variable

Formulas almost never arrive in the form you need. The area of a triangle is written , but a problem might give you the area and the height and want the base. Rewriting the formula so stands alone is called solving for a variable (or solving a literal equation), and it is the same isolate-the-variable process from Linear Equations — the only difference is that the numbers around your target variable are letters, so the answer is an expression instead of a single value.

Nothing about the algebra changes. Every move — add the same thing to both sides, multiply both sides by the same nonzero thing, distribute, combine like terms, factor — is exactly as it was before. What changes is that the “constants” you are working around are now written as letters, and the only genuinely new technique appears when the target variable shows up in more than one place. This lesson is a companion to Equations with More Than One Variable: it turns the routine into a decision checklist, then works case by case through fractions in the formula, the target appearing twice, the target in a denominator, and the target locked inside a power or a root.

The Solving-for-a-Variable Formula

Every other letter — here and — is treated exactly like a constant. The goal is to get the target letter alone on one side of the equals sign, with an expression made only of the remaining letters and numbers on the other. Both forms of the formula, and , say the same thing about how the quantities relate; you have just chosen a different letter to stand alone.

The Checklist

1. Name the target variable. Circle it in the formula. Every move from here is aimed at isolating that one letter, and nothing else.
2. Clear fractions. Multiply every term by the LCD of all denominators, or multiply both sides by the reciprocal of a lone fractional coefficient.
3. Distribute any parentheses that contain the target variable, so you can see every term it appears in.
4. Collect the target variable on one side, everything else on the other, using addition and subtraction.
5. If the target now appears in more than one term, factor it out so it appears exactly once.
6. Divide both sides by whatever is multiplying the target.
7. If the target is inside a power or a root, undo that last — take of both sides for a square, or square both sides for a square root.

Not every problem needs every step. A formula like skips straight to step 6. But every literal equation in this lesson is this checklist with one or two of the middle steps switched on.

Target Variable Appears Once

When the target shows up in exactly one term, the checklist finishes it directly: peel away everything attached to it by addition or subtraction, then divide by whatever multiplies it.

Notice that both and end up over the . Writing is a common slip — the division applies to the entire left side, not just its first term. Keeping the answer as one fraction, , makes that impossible to get wrong.

Target Variable Appears More Than Once

This is the one situation that needs a move you have not used before. Gather every term containing the target onto one side, factor the target out of those terms, then divide by whatever is left in the parentheses.

If the target starts on both sides of the equation, first move those terms together, then factor:

Factoring only works once the target is a common factor of every term on that side. Trying to divide by without factoring first is not a legal step.

Target Variable Inside a Fraction

If the target sits in a numerator, clearing the fraction is the usual multiply-by-the-denominator move. If it sits in a denominator, multiplying both sides by the target lifts it out, and then you isolate it normally.

Target Variable Inside a Power or a Root

Undo the outermost operation last. To free a variable that is squared, isolate the square, then take the square root of both sides:

To free a variable that is under a square root, isolate the root, then square both sides:

For , a physical length, only the positive root is kept. In a pure algebra context, where the variable could be negative, you would write . Squaring, like the square root, is applied to each entire side — , and .

Common Formulas Rearranged

Worked Example A: The Simple Interest Formula for t

Solve for .

Step 1 — the target is . It appears once, multiplied by and .

Step 2 — divide both sides by everything currently multiplying , which is :

Answer: .

Worked Example B: A Variable That Appears Twice

Solve for .

Step 1 — collect the -terms on the left, the constants on the right:

Step 2 — factor out of the left side:

Step 3 — divide by :

Answer: , valid as long as . Every had to be gathered onto one side before factoring — factoring is only possible once is a common factor of every term there.

Worked Example C: A Formula with a Fraction

Solve for .

Step 1 — multiply both sides by to clear the fraction:

Step 2 — subtract from both sides:

Answer: .

Worked Example D: Fahrenheit and Celsius

Solve for .

Step 1 — clear the fraction by multiplying both sides by :

Step 2 — add to both sides:

Answer: — the Celsius-to-Fahrenheit conversion, produced by solving the Fahrenheit-to-Celsius formula for the other letter.

Worked Example E: The Variable Under a Root

Solve for .

Step 1 — square both sides to remove the root:

Step 2 — divide by :

Answer: .

Worked Example F: Standard Form to Slope-Intercept Form

Solve for .

Step 1 — move the -term to the right side:

Step 2 — divide both sides by :

Answer: , which splits into — the slope-intercept form of the same line, with slope and -intercept .

Common Mistakes to Avoid

• Combining unlike letters. is not or ; and are not like terms just because they are both made of symbols.
• Dividing before factoring when the target appears twice. must become before you can divide by .
• Dividing only one term. In , both and are over the .
• Forgetting the when taking a square root in a context where the variable could be negative.
• Squaring only one term instead of the whole side. is ; is — square each entire side.
• Reversing which fraction gets used when clearing a fractional coefficient. Multiplying both sides by is the same as dividing by ; swapping them flips the formula.
• Losing track of the target. Decide which letter you are solving for before the first move, and keep aiming at it.

Where This Shows Up Later

• Systems of Equations. The substitution method starts by solving one equation for one variable in terms of the others.
• Functions and graphing. Rewriting a line from standard form to is a solve-for- problem, and it is what makes slope and intercept readable off the equation.
• Science and finance. Every formula in physics, chemistry, and finance eventually needs to be solved for a variable other than the one it is written for — exactly this skill, applied to a new formula each time.
• Rational and radical equations. Isolating a variable inside a fraction or a root is the same technique those lessons build on, with the added extraneous-solution check when an even power or root is involved.
• Word problems. Distance-rate-time, work, and mixture problems all use standard formulas (, ) rearranged for whichever quantity is unknown.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Solve for .

Problem 2. Solve for .

Problem 3. Solve for .

Problem 4. Solve for .

Problem 5. Solve for .

Problem 6. Solve for .

Problem 7. Solve for .

Problem 8. Solve for .

Problem 9. Solve for .

Problem 10. Solve for .

Problem 11. Solve for .

Problem 12. Solve for .

Problem 13. Solve for .

Problem 14. Solve for .

Quick Reference

This is the Linear Equations routine with letters for constants; the fraction-clearing step is Linear Equations with Fractions, and the deeper treatment with more literal-equation examples is Equations with More Than One Variable. More Algebra lessons are available too.