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Introduction
Rearranging Formulas helps you connect a mathematical rule to a result you can explain and check. Rearrange a measurement formula to make a specified variable the subject. Check the new formula with meaningful sample measurements. This lesson belongs to linear equations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in rearranging formulas.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Solve numerical linear equations and factor common terms.
Previous lesson: Literal Equations
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- expression
- A mathematical phrase representing a value.
- equation
- A statement that two expressions are equal.
- range
- The set of outputs produced.
- factor
- An expression multiplied by another to form a product.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Solve for a specified variable
Rearrange a measurement formula to make a specified variable the subject. Check the new formula with meaningful sample measurements.
A literal equation contains several letters. The target variable is the letter you must isolate; the other letters represent known or independently specified quantities. Apply inverse operations while keeping the expressions on both sides equal.
Undo outer addition or subtraction before undoing multiplication. If the target appears more than once, collect its terms and factor it out. Write parentheses around an entire numerator. Record restrictions whenever you divide by an expression that could equal zero.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. The target is w; treat P and l as known quantities.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Using l=3, w=3 gives P=129; rearranging returns w=3.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Solve P = 12l + 31w for w.
- The target is w; treat P and l as known quantities.
- Subtract 12l from both sides: P−12l=31w.
- Divide the ENTIRE left side by nonzero 31: w=(P−12l)/31.
Answer: w = (P − 12l)/31
Verify: Using l=3, w=3 gives P=129; rearranging returns w=3.
Avoid this mistake: Dividing only part of a sum: group the entire numerator.
Example 2 · Basic
Solve P = 13l + 31w for w.
- The target is w; treat P and l as known quantities.
- Subtract 13l from both sides: P−13l=31w.
- Divide the ENTIRE left side by nonzero 31: w=(P−13l)/31.
Answer: w = (P − 13l)/31
Verify: Using l=3, w=3 gives P=132; rearranging returns w=3.
Avoid this mistake: Treating other letters as zero: they still represent quantities.
Example 3 · Intermediate
Solve P = 14l + 31w for w.
- The target is w; treat P and l as known quantities.
- Subtract 14l from both sides: P−14l=31w.
- Divide the ENTIRE left side by nonzero 31: w=(P−14l)/31.
Answer: w = (P − 14l)/31
Verify: Using l=3, w=3 gives P=135; rearranging returns w=3.
Avoid this mistake: Forgetting restrictions on a variable denominator.
Example 4 · Advanced
Solve P = 2l + 32w for w.
- The target is w; treat P and l as known quantities.
- Subtract 2l from both sides: P−2l=32w.
- Divide the ENTIRE left side by nonzero 32: w=(P−2l)/32.
Answer: w = (P − 2l)/32
Verify: Using l=3, w=3 gives P=102; rearranging returns w=3.
Avoid this mistake: Dividing only part of a sum: group the entire numerator.
Example 5 · Challenge
Solve P = 3l + 32w for w.
- The target is w; treat P and l as known quantities.
- Subtract 3l from both sides: P−3l=32w.
- Divide the ENTIRE left side by nonzero 32: w=(P−3l)/32.
Answer: w = (P − 3l)/32
Verify: Using l=3, w=3 gives P=105; rearranging returns w=3.
Avoid this mistake: Treating other letters as zero: they still represent quantities.
Common mistakes and corrections
- Dividing only part of a sum: group the entire numerator.
- Treating other letters as zero: they still represent quantities.
- Forgetting restrictions on a variable denominator.
Real-world applications
- Rearrange a perimeter formula to find a missing side. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Solve a speed-distance-time relationship for a desired quantity. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Rearrange a measurement formula to make a specified variable the subject. Check the new formula with meaningful sample measurements.
A literal equation contains several letters. The target variable is the letter you must isolate; the other letters represent known or independently specified quantities. Apply inverse operations while keeping the expressions on both sides equal.
Frequently asked questions
What should I know before rearranging formulas?
Solve numerical linear equations and factor common terms.
How can I check my work?
Using l=3, w=3 gives P=129; rearranging returns w=3. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Dividing only part of a sum: group the entire numerator. Treating other letters as zero: they still represent quantities. Forgetting restrictions on a variable denominator.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Rearrange a perimeter formula to find a missing side.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Rearrange a measurement formula to make a specified variable the subject. Check the new formula with meaningful sample measurements.
- Dividing only part of a sum: group the entire numerator.
- Treating other letters as zero: they still represent quantities.
- Forgetting restrictions on a variable denominator.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
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Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials