# Equation Manipulation

Manipulating the equation by rearranging and making it easier to solve for the unknown. If you know how to do this then you can solve many problems.

We will make it real simple so that it is easy to understand. Remember, what ever you do on one side of the equation do the same on the other side

##### Example 1
```12 = a + 4 solving for a by getting a by itself on one side of the equation
to solve for a  you will have to cancel the + 4. you do that by subtracting 4  from a + 4
What ever you do on one side of the equation do the same on the other side and you end up with
12 - 4 = a + 4 - 4  -- the plus 4 and minus 4 cancel each other out and you end up with
12 - 4 = a```
##### Example 2
```12 = a - 4 here to isolate a you have to add a 4 to both  sides
12 + 4 = a - 4 + 4  which becomes 12 + 4 = a```
##### Example 3
```12 = a * 4 here to isolate the a divide both sides of the equation by 4
12 / 4 = a * 4 / 4  --- the multiply by 4 and divide by 4 cancels each other and it becomes 12 / 4 = a```
##### Example 4
```12 = a / 4  here to isolate the a multiply  both sides of the equation by 4
12 * 4 = a / 4 * 4 ---  the divide by 4 and then multiply by 4  cancels each other and it becomes
12 * 4 = a```
##### Example 5
```12 = a + b  here to isolate a you have subtract b from both sides
12 - b = a + b - b --- the plus and minus b cancel each other and the equation looks like this
12 - b =  a```
##### Example 6
```12 = a * b this time we isolate the b  to do that divide both sides by a
12 / a = a * b / a ---multiplying a * b and then dividing it by a  cancels each other and you are left with
12 / a = b```

If you understand all this then you should have little problem with solving sides and angles.

##### Example 7
Using the tangent function solving for an unknown side or angle in a triangle
tan A = opposite / adjacent
to solve something like this you need to know 2 of the values
If you know the angle and one side you can use the tangent function
A = 25°
b the adjacent side = 12 the equation would look like this
tan 25° = opposite / 12
You can get the value for tan 25° by looking it up in a natural tangent's chart or use a calculator set to degrees, punch in 25 and hit the tan button = 0.4663 rounded to 4 digits.

The equation then looks like this:
0.4663 = opposite / 12

To make it easier to solve rearrange the equation
```0.4663 =  opposite / 12 --- isolate opposite  by multiplying both sides by 12
0.4663 * 12  = opposite / 12 * 12 --- dividing and multiplying by 12 cancels each other and you end up with
0.4663 * 12 = opposite  ---   0.4663 * 12 = 5.5956
```
##### Example 8
Same problem as before but this time you know the 2 sides and need to solve the angle.
You can't use the tangent function. The tangent function takes an angle measurement as input and returns a ratio as the result. Arc tan takes a ratio as an input and returns an angle measurement as the result.
arctan A = opposite / adjacent
opposite = 6
arctan A = 6/12
arctan A = 0.5 on your calculator punch in the shift key, the up arrow, and 0.5 tan -1 and the result should look like 26.5651°

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