# Kinematics/Algebra Review

## Algebra Review

- Variables, Equations, Polynomials and Systems

``` A variable is a mathematical term represented by letters that can have varying values. If independent, the variable depends on no other variable(s). If dependent, the variable depends on other variable(s).
```

Imagine we are describing the location of a car. We could use two variables, x and y, to give its location in terms of longitude (x) and latitude (y). If we want to see how the car moves over the course of a day, we can include a variable for time, t. Then, measuring the longitude and latitude of the car over the day, we can express the position at a given time, x(t) and y(t). Here x and y depend on t, and thus are dependent variables. Time continues on its merry way, independent of any other variable.

``` An equation is a mathematical statement that shows two expressions to be the same, equal, to each other.
```
``` A linear equation is one in which the only power of the variable(s) is 1.
```
``` A homogeneous equation is one in which all variables of the same type {dependent, independent} are on one side of the equation whereas the other side is 0.
```

For example, x = 3 would mean that x is equal to, or x is, 3. x = y would mean that x is equal to y, or x is, y. A re-arrangement, x - y = 0, is an equation that is both homogeneous and linear. Whenever two expressions are equal, one may be directly substituted for the other in an equation.

``` An equation may be solved for a certain variable by means of manipulations conforming to the following laws:
Where k and c are constants,
If x - k = c, x = c + k
If x + k = c, k = c - k
If kx = c, x = c/k
If x/k = c, x = kc
```

The general rule to remember is that you perform the reverse operation on the other side of the equation.

An example is 5x + 3 = 96, you would first move 3 to the other side, using +'s reverse operator -: 5x = 96 - 3 = 93

And then divide by 5, using the reverse operator of multiplication, division: x = 93/5

``` A system is a set of equations containing 2 or more variables that have 1 solution, no solutions, or an infinite set of solutions.
```

The equations: x + y = 5 2x + 3y = 2 form a system of linear equations; solving them is a matter of using a substitution. (Though this is not the only method of solving them, other methods are beyond the scope of this book)

For example, let's use the substitution that y = 5 - x (From the first equation), we can then do the following: 2x + 3y = 2 2x + 3(5 - x) = 2

Expanding the bracket, by multiplying everything inside it by 3, yields: 2x + 15 - 3x = 2 -x + 15 = 2 -x = -13 x = 13

You now have one variable's solution; finding y's solution is using the initial substitution: y = 5 - x = 5 - 13 = -8 You now have the solution set (13, -8)

To be continued...