Rearranging Formulas and Literal Equations

2.5: Rearranging Formulas and Literal Equations. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 5 of 63 parts Practice Flashcards

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Isolate a numerator variable

Objective: Rearrange a formula by inverse operations.

Prerequisite skills: equation balance and distributive property. A literal equation contains multiple letters, but the operations are unchanged. Treat every other letter as a known quantity and isolate the requested letter. State the condition on any divisor introduced. Common error: dividing y by m before subtracting b. Restrictions and interpretation: Record nonzero denominators and coefficients before division.

Formulas

$y = mx + b ⇒ x = (y − b)/m, m ≠ 0$

Worked examples

Try itIn y = mx + b, y = 514, m = 23, and b = 8. What is x?Show answer

Subtract b to obtain y − b = mx. Divide by m, which is nonzero: x = (514 − 8)/23 = 22. Therefore the requested answer is 22.

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Isolate a numerator variable

Objective: Rearrange a formula by inverse operations.

Formulas

$y = mx + b ⇒ x = (y − b)/m, m ≠ 0$

Worked examples

Try itIn y = mx + b, y = 514, m = 23, and b = 8. What is x?Show answer

Subtract b to obtain y − b = mx. Divide by m, which is nonzero: x = (514 − 8)/23 = 22. Therefore the requested answer is 22.

Formulas with products

Objective: Solve a product formula for a selected factor.

When the unknown multiplies known factors, divide by their entire product. Geometric and rate formulas often have a numerical coefficient such as 1/2; multiplying by its reciprocal removes it. Common error: forgetting to undo the factor 1/2. Restrictions and interpretation: Record nonzero denominators and coefficients before division.

Formulas

$A = bh/2 ⇒ h = 2A/b, b ≠ 0$

Worked examples

Try itA triangle has area 325 square units and base 26 units. Using A = bh/2, what is its height?Show answer

Multiply A = bh/2 by 2: 2A = bh. Divide by b = 26: h = 2(325)/26 = 25 units. Therefore the requested answer is 25.

Unknown in several terms

Objective: Factor the requested variable before dividing.

If a letter occurs in several terms, collect those terms and factor out that letter. Divide by the remaining factor only if it is nonzero. If that factor is zero, return to the original equation to classify the case. Common error: dividing by just one coefficient when the variable has two terms. Restrictions and interpretation: Record nonzero denominators and coefficients before division. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$px + qx = r ⇒ x = r/(p + q), p + q ≠ 0$

Worked examples

Try itThe formula px + qx = r has p = 25, q = 3, and r = 756. What is x?Show answer

Factor x: x(p + q) = r. Here p + q = 28 is nonzero. Divide: x = 756/28 = 27. Therefore the requested answer is 27.

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