Rearrange Formulas to Solve for Any Variable

Lesson 1 of 1

In this lesson:

• Rearranging a formula IS equation solving — same moves, same logic
• Apply the strategy to geometry, physics, chemistry, and finance formulas
• Recognize multi-step cases and choose the right form for each problem

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Isolate any variable in a formula using inverse operations
2. Treat other variables as constants while solving
3. Apply rearrangement to physics, geometry, and finance formulas
4. Choose the most useful form of a formula

Every Formula Hides Multiple Relationships

From (Ohm's Law), rearrange for any variable:

• Find :
• Find :
• Find :

One formula — three tools

Rearranging Formulas Is Equation Solving

The steps are identical — letters play the role of "known" quantities:

• In : the numbers 2, 3, 11 are given
• In : the variables and are treated as given (constants)

Two Examples Compared Side by Side

Numeric equation:

Formula rearrangement (same moves):

Same two inverse operations — same logic as the numeric case

Quick Check: The Equation-Formula Connection

For , solving for :

Why can we treat and as constants while solving for ?

Think: what would the numeric equivalent look like?

Answers to Five Formula Rearrangements

Apply inverse operations to isolate the target

Practice with One-Step Formula Rearrangements

Undo one operation to isolate the target variable:

Rule: Last operation applied → first operation undone

Two-Step Rearrangements: Geometry and Finance

Divide or multiply first, then add or subtract — as needed

Rearranging Formulas in Science Contexts

Treat , , , , , as constants — divide both sides by the product

Guided Practice: One- and Two-Step

Rearrange each formula for the indicated variable:

1. — solve for
2. — solve for
3. — solve for

Write the rearranged formula before checking the answer

Answers: One- and Two-Step Practice

Each answer is a formula itself — use it for any values of the other variables

Practice Set: Rearranging Common Formulas

Rearrange each formula for the indicated variable:

1. — solve for
2. — solve for
3. — solve for
4. — solve for (confirm your earlier result)

Answers to Formula Rearrangement Practice

Multi-Step: Target Variable Appears Twice

When the target appears in multiple terms, factor it out first:

1. appears in two terms — factor:

2. Divide both sides by :

Multi-Step: Target Inside a Square Root

1. Isolate :

2. Take the square root ():

Multi-Step Rearrangements with Roots and Powers

Take the square root last — after isolating

Quick Check: Planning a Multi-Step Strategy

For , a student writes:

Is this step correct? Explain.

Then: What should the student do next?

Guided Practice: Multi-Step Formula Rearrangement

Solve for :

Hint: Isolate first by moving the other term.

Write each step before the answer

Multi-Step Formula Rearrangement Practice Problems

Rearrange each for the indicated variable:

1. — solve for
2. — solve for
3. — solve for (assume )

Answers to Multi-Step Practice Problems

Choosing the Right Form of a Formula

Ask: What do you know? What do you need?

Application: Three Problems, One Strategy

Problem 1: A 100 m race completed in 9.58 s. Find average speed.

Problem 2: Cylinder: cm³, cm. Find height .

Application: Solving an Interest Rate Problem

Problem: 500 grew to 650 in 3 years (simple interest). Find annual rate .

Units: dollars / (dollars · years) = 1/year — rate is per year ✓

Guided Practice: Choose and Apply

For each problem, identify the formula, rearrange it, then substitute:

1. A 400-meter swim completed in 250 seconds. Find average speed.

2. A simple-interest loan: $800 borrowed, $72 interest, rate = 6%. Find the time.

Identify which variable to solve for before computing

Key Takeaways from This Lesson

✓ Rearranging = equation solving with letters as constants

✓ Undo operations in reverse order (last applied → first undone)

Target in multiple terms? Factor it out before dividing

Target squared? Isolate the power, then take the root last

Apply every operation to the entire other side

What Comes Next: Equivalent Expressions

This skill appears across all of math and science:

• Physics: ,
• Chemistry: ,
• Finance:

Rearrange once — reuse the form for every future problem.