How to manipulate formulas Algebra Crash Course How to manipulate formulas

What exactly is algebra…to a scientist? For scientists, algebra is a tool used for solving mathematical problems. It helps us not have to memorize as many formulas…we learn one and then can manipulate it to solve for what we want.

How does it work? We start off with a problem to solve… 3x + 7y = 24 Solve for x The words “solve for” or “find” mean we want to get “x” by itself on one side of the equation.

Manipulation In order to get something by its self, we have to manipulate the equation… Manipulate simply means to move around or rearrange… So we have to rearrange the equation.

3x + 7y = 24 To solve for x, we have to identify all of the “terms” that have an x in them and all that do not. 3x has an x 7y and 24 do not… So…

3x + 7y = 24 We want to move ALL of the TERMS that do not have x to one side and all of the terms that do have x to the other side… So we need to move the 7y to the same side as the 24

Moving things… To move something from one side of the equation to the other, we have to “do the opposite” This means that we have to perform the opposite operation than what is being done to it right now…

3x + 7y = 24 The 7y is being added to the 3x, so we need to subtract it 3x + 7y = 24 - 7y This cancels out the 7y on the left…but we’re not done yet. 3x = 24

What’s done to one is done to the other. What ever you do to one side of the equation, you MUST do to the other side… Since we subtracted 7y from the left side, we have to subtract it from the right side… 3x = 24 – 7y

Continue the process We must continue this until the thing we want is by itself on one side… 3x = 24 – 7y X is not by itself…is multiplied by 3, so we need to divide by three…

Continue the process 3x = 24 – 7 3 This cancels out the 3 on the left…but what is done to one side must be done to the other…so x = 24 – 7y

Now we need to simplify if possible x = 24 – 7y 3 24 / 3 = 8 so…. x = 8 – 7y That’s the answer…we are done…

Let’s do some examples…. 3x2 + zyw – y2 = z – 4; solve for x First we need to identify the different terms… Now move all terms without an x to one side…

Example 3x2 + zyw – y2 = z – 4 -zyw + y2 -zyw + y2 To get… 3x2 = z – 4 - zyw + y2 Now divide by 3…to get x2 = z – 4 - zyw + y2 3

Example x2 = z – 4 - zyw + y2 3 Now we have to get rid of the “squared”…so we do the opposite…what is the opposite of squaring something? The square root!

Example x2 = z – 4 - zyw + y2 3 This gets rid of the squared to give…

Example x = z – 4 - zyw + y2 3 That’s our answer

Letters… As you can see from the previous example…you don’t need to have numbers to do algebra… As you will see in many physics problems, plugging in numbers will actually make things more difficult. You need to learn to perform algebra with letters as well as numbers.