Solving for a Variable

Lesson Plan - Get It!

Audio:

Formulas are a major part of the sciences, and there is one question.

• How do we solve for specific variables?

• How do we take a formula such as the formula for distance traveled, distance = rate * time, and solve for just one of the variables, 'rate'?"

(OK, that's two questions.)

When you work in the sciences, many times you will only be able to find out parts of truths from your experiments, and by rearranging formulas and working with the knowledge that you do have, you can determine what you want to know and get all the information.

Often, you also must convert between multiple units, usually between standard measurements (in the U.S.) and the metric system, but there are many more formulas that will help you find relationships between different quantities or facts.

Remember to always write the units when working with formulas! Otherwise, you won't remember what the numbers mean!

For an example, take a look at the 2016 Olympics:

The winning time for the men's marathon was from Eliud Kipchoge of Kenya, at 2 hours, 8 minutes, 44 seconds or, in decimal form, 2.1456 hours.

If you want to find out on average how fast he ran each of the 26.2 miles, then you take the distance formula and divide both sides by time to get rate = ^distance⁄_time  or  ^distance⁄_time = rate depending on how you prefer to write equations.

Now, you can just substitute in what we know to find out what his average speed was:

^26.2⁄_2.1456 = average speed

Which, when you plug into a calculator, equals an average speed of 10.7 miles per hour, which is pretty fast!

If we want to find out another part of the distance formula — say the time it took to run a marathon — then we can take an average speed of 10 miles an hour and a distance of 26.2 and set it up in an equation as either 26.2 = 10t, where t is time in hours, or we could solve for our desired variable time first and get the equation  ^distance⁄_{average speed} = time.

Then, we don't have to worry about mixing our numbers up while we are solving, and can just plug them right in for time = ^26.2⁄₁₀ or a running time of 2.62 hours.

Another example would be trying to convert between Fahrenheit and Celsius, given by the formula F = ⁹⁄₅ C + 32.

• This is a great formula to use if we have measurements in Celsius and we are trying to get them into degrees Fahrenheit, but what if we are going the other way?

Say we want to convert degrees Fahrenheit into degrees Celsius; then, we have to solve for C by subtracting 32 from both sides to get  F - 32 = ⁹⁄₅ C then we can multiply the fraction by its inverse to get C = ⁹⁄₅ (F - 32).

So, if we have a Fahrenheit reading of 86, then when we plug that into our formula, we get C = ⁹⁄₅ (54) which equals 30 degrees Celsius.

If we want to go from Celsius of 40 degrees to Fahrenheit, then we take the Fahrenheit formula and plug in the Celsius reading to get F = ⁹⁄₅ (40) + 32 which equals 104 degrees Fahrenheit.

Knowing this, we can solve any formula for any variable.

If you are having difficulty check out this solve for indicated variable video from Sandra Ofili:

To make sure you have understood the main ideas of this lesson, go on to the Got It? section for a bit of practice.