Math You Need > Rearranging Equations

How do I isolate x (or P or T...) in a formula?
Rearranging equations to solve for a given variable

Equations as important geological tools

A professor speaking "Math", which can seem like another language! photo by Jennifer M. Wenner.
Sometimes, it may seem like your geoscience instructor is speaking another language when he/she talks about equations or formulae. Especially if he/she expects you to "manipulate" or rearrange them! But, equations can provide powerful tools for describing the natural world. In the geosciences, we can describe the behavior of many natural phenomena by writing an equation for a line (y = mx + b), or with exponential functions (y = ext). And with a little algebra, we can rearrange those equations to solve for ANY of the variables in them.

Although this may seem like magic, you don't have to be a "mathemagician" to do this. This page is designed to give you some tools to call upon to help you to learn some simple steps to help you to solve an equation for any of the variables (letters that represent the element or quantity of interest).

Why should I manipulate equations?

juggling math operations
Photo of Ken Andrews (scientist at JPL) juggling. Modified by Jen Wenner.

Believe it or not, there are many good reasons to develop your ability to rearrange equations that are important to the geosciences. It can save time, help you with units and save some brain space! Here are some reasons to develop your equation manipulation skills (in no particular order):

Where is this used in the geosciences?

To be honest, equation manipulation occurs in almost every aspect of the geosciences. Any time you see a P or T or ρ or x (or even =), there is an equation that you could manipulate. Because equations can be used to describe lots of important natural phenomena, being able to manipulate them gives you a powerful tool for understanding the world around you!

See the Practice Manipulating Equations page for just a few examples.

A Review of Important Rules for Rearranging Equations

Solving for y by Jennifer M. Wenner.

You probably learned a number of rules for manipulating equations in a previous algebra course. It never hurts to remind ourselves of the rules. So let's review:

Some simple steps for manipulating equations

Here are some simple steps for manipulating equations. Under each step you will find an example of how to do this with an example that uses the geologic context of density (a measure of mass per unit volume).

  1. Assess what you have (which of the variables do you have values for?, what units are present?, etc.). DO NOT plug in any numbers yet!
    A pyrite cube For example: You have a cube of pyrite that is 3 cm x 3 cm x 3 cm. You know that pyrite's density is 5.02 g/cm3. Can you figure out how much that cube of pyrite weighs (without using a balance)?

    First, you need to know that density (ρ) is equal to mass (m) divided by volume (v). We can write this as a mathematical expression (or equation, if you prefer):

    equation for density (variables)

    Which of these values do you have in the question above? You have density (5.02 g/cm3). And with the information you can figure out volume (length x width x height).
  2. Determine which of the variables you want as your answer. (What is the question asking you to calculate? What is the unknown variable?)
    The question above asks you to determine the mass of a pyrite cube (without weighing it/using the information given in the problem). So, in the equation for density, you want to determine "mass". Remember, don't plug anything in yet.
  3. Rearrange the equation so that the unknown variable is by itself on one side of the equals sign (=) and all the other variables are on the other side. RULE #1: you can add, subtract, multiply and divide by anything, as long as you do the same thing to both sides of the equals sign.
    Let's take the density equation:

    equation for density (variables)

    and rearrange it. We want to isolate the variable for mass (m). To do this we first multiply both sides of the equation by volume (v).
    isolating mass in the density equation
    Then, we can cancel volume on the right side of the equation (volume รท volume = 1).
    cancel volume to solve for mass

    Note that these first two steps are the same as cross-multiplying. If you are more familiar with this method, you can do that as well. Either way...
    We end up with an equation that has mass isolated on one side of the equation!
    solving for mass
  4. NOW plug in the numbers! Replace known variables with their values and don't forget to keep track of units!
    Our equation is solving for mass . The nice thing about this equation is that now that we've rearranged it, all of our known variables are on one side and the one we don't know is on the other. Begin by plugging in what we know: ρ (the density of pyrite) and V (the volume (length x width x height) of the cube):

    plugging in density and l x w x h

    Simplify the volume term by multiplying:

    plugging in density and calculated volume

    Cancel same units on the top and bottom (where you can) so that we end up with the units we want (if you don't understand how to do this, see the Unit Conversions module):

    cancel volume units to solve for mass
  5. Determine the value of the unknown variable by performing the mathematical functions. That is, add, subtract, multiply and divide according to the equation you wrote for step 2.
    In this case, it is a simple multiplication:

    cancel volume units to solve for mass

    And we end up with mass:

    calculated mass of pyrite cube
  6. Ask yourself whether the answer is reasonable in the context of what you know about the geosciences and how much things should weigh.
    student holding large piece of pumice

    This is a thing that mostly takes experience. If you are unsure, you could find a balance and weigh the cube to see if you're in the right ballpark. If you're holding it in your hand, you could guess whether this seems about right...More importantly, if you get a number like 135,000 g, do you think that's reasonable? That's 135 kg (which is about 300 lbs!) and it is probably not right. What about if you get something like 0.00135 grams? It is important to be able to distinguish whether you're in the right range, more than whether you're exactly right.

    Another way to think about whether you're right is to find something that weighs the same from your own experience. What does 135 g feel like? Well, there are about 450 g in a pound, so 135 g is between 1/4 lb and 1/3 lb. What do you know that has a similar weight? (The first think that comes to mind for me is burgers...). Does it make sense that a cube of pyrite (a golden metallic mineral) that is about one inch on each side would weigh that much? Use your own experience to develop a way to evaluate weights and other measures.

Next Steps

More help with equations

Geomaths at University College London has a MathHelp page about equations and functions (more info) .

The chemistry department at Texas A&M has a math review page about Algebraic Manipulation.

The Economics and Business faculty at University of Sidney has a page where you can practice your equation manipulation skills! Take the algebraic manipulation quizzes!
This page was written and compiled by Dr. Jennifer M. Wenner, Geology Department, University of Wisconsin Oshkosh
and Dr. Eric M. Baer, Geology Program, Highline Community College,

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