4251 3

0011 0010 1010 1101 0001 0100 1011

Algebra Crash Course

How to manipulate formulas

4251 3

0011 0010 1010 1101 0001 0100 1011

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.

4251 3

0011 0010 1010 1101 0001 0100 1011

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 its self on one side of the equation.

4251 3

0011 0010 1010 1101 0001 0100 1011

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.

4251 3

0011 0010 1010 1101 0001 0100 1011

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…

4251 3

0011 0010 1010 1101 0001 0100 1011

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

4251 3

0011 0010 1010 1101 0001 0100 1011

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…

4251 3

0011 0010 1010 1101 0001 0100 1011

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

4251 3

0011 0010 1010 1101 0001 0100 1011

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

4251 3

0011 0010 1010 1101 0001 0100 1011

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…

4251 3

0011 0010 1010 1101 0001 0100 1011

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

3

4251 3

0011 0010 1010 1101 0001 0100 1011

Now we need to simplify if possible

x = 24 – 7y

3

24 / 3 = 8 so….

x = 8 – 7y

3• That’s the answer…we are done…

4251 3

0011 0010 1010 1101 0001 0100 1011

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…

4251 3

0011 0010 1010 1101 0001 0100 1011

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

4251 3

0011 0010 1010 1101 0001 0100 1011

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!

4251 3

0011 0010 1010 1101 0001 0100 1011

Example

x2 = z – 4 - zyw + y2

3• This gets rid of the squared to give…

4251 3

0011 0010 1010 1101 0001 0100 1011

Example

x = z – 4 - zyw + y2

3• That’s our answer

4251 3

0011 0010 1010 1101 0001 0100 1011

Letters…

• As you can see from the previous example…you don’t have 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.