Isolate a Quantity in a Multi-Variable Formula
Rearranging a formula uses the same balanced operations as solving a numerical equation.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Rearrange A=(h/2)(p+q) to solve for q, with h≠0. Candidate answers: (A) q=2A/(h−p) (B) q=2A/h−p (C) q=2Ah−p (D) q=A/(2h)−p.
- Independent check inputs: Rearrange C=2πr to solve for r. Candidate answers: (A) r=2πC (B) r=C/(2π) (C) r=C/2 (D) r=πC/2.
Learn the idea
An equation may describe distance, a bounded interval, a parameter-dependent relation or a formula between several quantities. Its solution must respect all stated conditions, including endpoint inclusion and nonzero divisors. Rearranging a formula uses the same balanced operations as solving a numerical equation. Treat other quantities as given symbols and explicitly state any required nonzero divisor.
Here is the method on the supplied teaching case. Multiply both sides by 2/h, then subtract p. The condition h≠0 permits the division.
Parentheses determine which entire group is multiplied. Invert that group operation on both sides before removing an added quantity; the result must be checked by substituting it back into the formula.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Rearrange A=(h/2)(p+q) to solve for q, with h≠0. Candidate answers: (A) q=2A/(h−p) (B) q=2A/h−p (C) q=2Ah−p (D) q=A/(2h)−p. Solution: Multiply both sides by 2/h, then subtract p. The condition h≠0 permits the division. The final answer is q=2A/h−p. Parentheses determine which entire group is multiplied. Invert that group operation on both sides before removing an added quantity; the result must be checked by substituting it back into the formula.
Check your reasoning
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Helpful tips
- Rearranging a formula uses the same balanced operations as solving a numerical equation.
- Parentheses determine which entire group is multiplied.
See the answer and explanation
r=C/(2π)
Divide the entire equation by nonzero 2π. Multiplying the proposed result by 2π reconstructs C.
Take it further
Changed-input task: Rearrange y=kx+b to solve for b. Candidate answers: (A) b=(y−k)x (B) b=y+kx (C) b=y/k−x (D) b=y−kx. Show the mathematical reason, not only the selected answer. Expected result: b=y−kx. Subtract the entire kx term from both sides. No division is required, so k=0 remains allowed for this rearrangement.
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More practice: try new cases
Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.
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Check the subject sources
- OpenStax: Linear inequalities and absolute value inequalities
- OpenStax: Linear equations in one variable
These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.