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...