College : Elimination Method for Solving a System of Equations - Math Problem Example

The "Blog"-version:9:30I was puzzled to find out that the vectors connecting the same two points (let's say QR and RQ) are not equivalent.

How can it happen It seems so logical to assume that if two endpoints are the same, the segment between them is the same also. Can somebody help me with this issue 9:50 Hi! It is simple, though unusual. Vectors differ from many other mathematical notions because a vector is determined by both a numerical value and the vector's direction. So, the vector is not only a number. You can imagine a vector as an arrow of a certain length.10:13 So, if the same point is the start for one vector and the ending point for another one and vice versa, even if these two vectors have the same length, they would not be equivalent10:28Exactly!

You can compare it with buses that have the same route but go in the opposite direction. If you stand at a bus stop waiting for the bus to go to the nearby town, it does not help you if the same minute the bus leaves the town of your destination and heads the other way. Although it is the same bus, you can not ride it as it goes in a different direction. The straight-line movement of the bus can be characterized by a vector. 11:05 Thanks a lot!

If the coefficients before one of the variables in two of the equations are the same, but have the opposite sign (or have the same sign - then one of the equations should be simply multiplied by -1) then it would be more practical to use the elimination method. This method is also handy for solving bigger systems that contain three or more variables. If in one of the equations, the coefficients of some of the variables are multiples of the corresponding coefficients in another equation, the elimination method allows significant simplification of the original system.

A good sign that the substitution method of solving a system of equations would be appropriate to use is the fact that in one of the equations, one of the variables stands alone on one side of the equation. However, the fact that it happens not so often does not mean that the substitution method is used rarely. An equation can be transformed into such a form. If such a transformation is easy and one of the variables can be quickly expressed as equal to some combination of other variables and numbers, the substitution method is the right choice to be made. Easy transformations are usually possible when one of the variables has a coefficient of +/- 1 or the coefficients of most other variables are multiples of it or if several variables have a coefficient of 0.