Mathovia

Math Practice

Solving for a Variable Practice Problems

Fifteen hand-built problems on solving a formula for a specified variable, with complete solutions. Each shows the target variable circled, the moves that isolate it, and the final rearranged formula.

  1. 1.

    Solve \(d = rt\) for \(t\).

    Show full solution

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

  2. 2.

    Solve \(A = \dfrac{1}{2}bh\) for \(b\).

    Show full solution

    $$ 2A = bh \quad\Longrightarrow\quad b = \frac{2A}{h} $$

  3. 3.

    Solve \(3x - 4y = 24\) for \(y\).

    Show full solution

    $$ -4y = 24 - 3x \quad\Longrightarrow\quad y = \frac{3x - 24}{4} $$

  4. 4.

    Solve \(P = 2\ell + 2w\) for \(w\).

    Show full solution

    $$ P - 2\ell = 2w \quad\Longrightarrow\quad w = \frac{P - 2\ell}{2} $$

  5. 5.

    Solve \(y = mx + b\) for \(m\).

    Show full solution

    $$ y - b = mx \quad\Longrightarrow\quad m = \frac{y - b}{x} $$

  6. 6.

    Solve \(V = \dfrac{1}{3}\pi r^{2}h\) for \(h\).

    Show full solution

    $$ 3V = \pi r^{2}h \quad\Longrightarrow\quad h = \frac{3V}{\pi r^{2}} $$

  7. 7.

    Solve \(F = \dfrac{9}{5}C + 32\) for \(C\).

    Show full solution

    $$ F - 32 = \frac{9}{5}C \quad\Longrightarrow\quad C = \frac{5}{9}(F - 32) $$

  8. 8.

    Solve \(y = \dfrac{x - 3}{4}\) for \(x\).

    Show full solution

    $$ 4y = x - 3 \quad\Longrightarrow\quad x = 4y + 3 $$

  9. 9.

    Solve \(A = P + Prt\) for \(t\).

    Show full solution

    $$ A - P = Prt \quad\Longrightarrow\quad t = \frac{A - P}{Pr} $$

  10. 10.

    Solve \(E = mc^{2}\) for \(c\).

    Show full solution

    $$ c^{2} = \frac{E}{m} \quad\Longrightarrow\quad c = \sqrt{\frac{E}{m}} $$

    (or \(\pm\sqrt{\ }\) in a pure algebra context.)

  11. 11.

    Solve \(ax + b = cx - d\) for \(x\).

    Show full solution

    $$ ax - cx = -d - b \quad\Longrightarrow\quad x(a - c) = -d - b \quad\Longrightarrow\quad x = \frac{-d - b}{a - c} $$

  12. 12.

    Solve \(S = 2\pi r h + 2\pi r^{2}\) for \(h\).

    Show full solution

    $$ S - 2\pi r^{2} = 2\pi r h \quad\Longrightarrow\quad h = \frac{S - 2\pi r^{2}}{2\pi r} $$

  13. 13.

    Solve \(m = rx + sx\) for \(x\).

    Show full solution

    $$ m = x(r + s) \quad\Longrightarrow\quad x = \frac{m}{r + s} $$

  14. 14.

    Solve \(d = \sqrt{x + 7}\) for \(x\).

    Show full solution

    $$ d^{2} = x + 7 \quad\Longrightarrow\quad x = d^{2} - 7 $$

  15. 15.

    Solve \(\dfrac{1}{f} = \dfrac{1}{a} + \dfrac{1}{b}\) for \(f\).

    Show full solution

    $$ \frac{1}{f} = \frac{a + b}{ab} \quad\Longrightarrow\quad f = \frac{ab}{a + b} $$
M
Written by
Mathovia Team
Editorial Team

Solving for a Variable Practice Problems

The Solving for a Variable lesson turns the isolate-the-variable routine into a checklist; this set is 15 formulas to rearrange, each with a full solution.

How to use this set

These problems are hand-written, not auto-generated. For each formula, decide which letter is the target, then apply the checklist: clear fractions, distribute, collect the target on one side, factor if it appears twice, divide, and undo any power or root last. Then compare with the solution.

Common mistakes to watch for

Combining unlike letters. is not or .

Dividing before factoring when the target appears twice.

Dividing only one term of a numerator by the divisor.

Forgetting the when taking a square root, or squaring only one term instead of the whole side.

Where to go next

Clearing fractions inside a formula is the Linear Equations with Fractions Practice Problems skill. For the deeper literal-equation treatment, see the Solving for a Variable lesson and Equations with More Than One Variable.

Frequently Asked Questions

How do I solve a formula for a variable when the other values are unknown?+

Treat every letter except the target as a constant. You can add, subtract, multiply, divide, or factor exactly as with numbers; the answer is an expression instead of a single value.

What do I do when the target variable appears in two terms?+

Move every term containing it to one side, factor the variable out, and divide both sides by whatever is left in the parentheses. Factoring is the one genuinely new step here.

How do I isolate a variable that is squared or under a root?+

Undo the outer operation last. Squared: isolate the square, then take of both sides. Under a root: isolate the root, then square both sides.

Do I clear fractions the same way?+

Yes — multiply every term by the LCD, or multiply both sides by the reciprocal of a single fractional coefficient.

When do I worry about dividing by zero?+

Whenever you divide both sides by an expression containing a variable, that expression is assumed nonzero. Most formulas guarantee this; note it when the divisor could plausibly be zero.

Why is this a curated set rather than auto-generated?+

The answers are algebraic expressions, not single numbers, so an automatic answer-checker can't verify them. Each problem has a full worked solution instead.

More practice problems

Linear Equations with Fractions Practice Problems Algebra Equation Practice Problems Quadratic Equations Practice Problems