Isolating Quantities: Foundations

[Video: Introduction to Isolating Variables]

Learn essential techniques for isolating variables and quantities in equations, a fundamental skill for solving SAT math problems.

Understanding Isolating Quantities

Isolating a quantity means rearranging an equation to get a specific variable or expression by itself on one side.

Why Isolate Quantities?

Solve for unknown variables
Rewrite formulas for different purposes
Make equations easier to interpret
Prepare equations for graphing

Basic Isolation Techniques

Step-by-Step Approach

1. Identify the quantity you want to isolate

2. Undo operations in reverse order of operations (PEMDAS)

3. Perform the same operation to both sides

4. Simplify at each step

5. Check your solution

Example: Isolate x in 2x + 5 = 13

1. Subtract 5 from both sides: 2x = 8

2. Divide both sides by 2: x = 4

Working with Formulas

Common Formula Rearrangements

Many SAT problems require rearranging common formulas:

Example 1: Solve for r in A = πr²

1. Divide both sides by π: A/π = r²

2. Take square root of both sides: r = √(A/π)

Example 2: Solve for b in a² + b² = c²

1. Subtract a² from both sides: b² = c² - a²

2. Take square root: b = ±√(c² - a²)

Advanced Techniques

When Variables Appear Multiple Times

For equations where the target variable appears in multiple terms:

Gather all terms with the variable on one side
Factor out the variable
Divide by the remaining expression

Example: Solve for y in 3xy + 2y = 10

1. Factor y: y(3x + 2) = 10

2. Divide both sides by (3x + 2): y = 10/(3x + 2)

Key Takeaways

Always perform the same operation to both sides
Work in reverse order of operations (PEMDAS backwards)
For fractions, cross-multiply to eliminate denominators
When variables appear multiple times, factor them out
Check your work by plugging the solution back in

Practice Question

The formula for the surface area of a cylinder is given by S = 2πr² + 2πrh, where r is the radius and h is the height. Which of the following equations correctly expresses h in terms of S and r?

A) h = (S - 2πr²)/(2πr)

B) h = (S - πr²)/(2πr)

C) h = (S)/(2πr) - r

D) h = (S - 2πr)/(2πr²)

Follow these steps:

Start with the original equation: S = 2πr² + 2πrh
Isolate the term containing h (2πrh)
Divide by the remaining factors to solve for h
Check your answer by comparing with the options

Step-by-Step Solution:

1. Start with: S = 2πr² + 2πrh

2. Subtract 2πr² from both sides: S - 2πr² = 2πrh

3. Divide both sides by 2πr: (S - 2πr²)/(2πr) = h

4. Rewrite: h = (S - 2πr²)/(2πr)

The correct answer is A) h = (S - 2πr²)/(2πr).

Note: Option C is algebraically equivalent but not in the same simplified form.