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## Digital SAT Math

### Unit 12: Lesson 6

Isolating quantities: advanced# Isolating quantities | Lesson

A guide to isolating quantities on the digital SAT

## What are isolating quantities problems?

Many real-world scenarios can be described using equations and formulas with multiple variables. For example, the formula for the area, A, of a rectangle with length ell and width w is A, equals, ell, w. In this form, area (A) is

**isolated**: it is alone on one side of the equation.If we want to find the equivalent equation, but with ell

**isolated**, we can divide both sides of the equation by w, giving us ell, equals, start fraction, A, divided by, w, end fraction.**You can learn anything. Let's do this!**

## How do I isolate quantities?

### Manipulating formulas: temperature

### Like solving equations, but with more variables

Isolating quantities is similar to solving equations. However, instead of ending up with a variable equal to a constant, we end up with a variable equal to an expression containing other variables. In fact, if you've rewritten a linear equation like x, plus, y, equals, 1 in slope-intercept form, then you've already isolated y by rearranging a linear equation!

The most important thing to remember is that the rearranged equation will remain equivalent to the original equation

*only if*we always treat both sides equally: whenever we do something to one side, we*must*do the exact same thing to the other side.To isolate a quantity in an equation or formula:

- Write down the original equation. If needed, translate the word problem or given context into an equation.
- Perform the same operation on both sides of the equation to begin isolating the desired quantity.
- Repeat Step 2 until the desired quantity is isolated.

#### Let's look at some examples!

A physics student uses the formula E, start subscript, start text, k, end text, end subscript, equals, start fraction, 1, divided by, 2, end fraction, m, v, squared to calculate the kinetic energy in joules, E, start subscript, start text, k, end text, end subscript, of an object with a mass of m kilograms traveling at a speed of v meters per second. What is v in terms of E, start subscript, start text, k, end text, end subscript and m ?

The distance d traveled by an object moving at constant velocity is found by multiplying the velocity v of the subject by time t. Write an equation that gives the time t in terms of d and v.

### Try it!

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