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# Isolating quantities | Lesson

## What is "Isolating quantities" and how frequently do these questions appear on the test?

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 wanted to find the equivalent equation, but with ell isolated, we could divide both sides of the equation by w, giving us ell, equals, start fraction, A, divided by, w, end fraction.
On your official SAT, you'll likely see 0 to 2 questions that test your ability to isolate a variable in an equation with two or more variables.
You can learn anything. Let's do this!

## How do I isolate quantities?

### Manipulating formulas: temperature

Manipulating formulas: temperatureSee video transcript

### 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:
1. Write down the original equation. If needed, translate the word problem or given context into an equation.
2. Perform the same operation on both sides of the equation to begin isolating the desired quantity.
3. 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!

try: identify the steps to isolate a variable
V, equals, pi, r, squared, h
The equation above shows the volume V of a right cylinder with radius r and height h.
To rewrite the equation to isolate r, we must first isolate r, squared. To do so, we
both sides of the equation by pi, h.
Next, to isolate r, we
both sides of the equation.
Which of the following correctly expresses r ?

practice: isolate a quantity in one step
Giselle uses the formula I, equals, P, r, t to calculate I, the simple interest in dollars, she collects from a loan of P dollars at an interest rate of r per month over a period of t months. Which of the following expresses P in terms of the other variables?

practice: write an equation, then isolate a quantity
The perimeter P of a rectangle is found by multiplying the sum of the rectangle's length l and width w by 2. Which of the following equations gives the width w in terms of P and l ?

practice: isolate a quantity with exponents
F, equals, start fraction, G, m, start subscript, 1, end subscript, m, start subscript, 2, end subscript, divided by, d, squared, end fraction
The equation above shows the F, the gravitational force between two objects, in terms of G, the gravitational constant, m, start subscript, 1, end subscript and m, start subscript, 2, end subscript, the masses of the two objects, and d, the distance between the two objects. Which of the following correctly expresses d in terms of F, G, m, start subscript, 1, end subscript, and m, start subscript, 2, end subscript ?