Or making friends with it
long has been algebra the reason of slaughter of morale for test-takers . Today we Declare holy war against the tyranny of oppression. Unlike uncle Sam our policy will be to befriend it and love it and thus conquer it and make it your ally for future battles of thought .
This is the Algebra Super post. The agenda will be to keep each idea in itself as short and simple and build a series of analysis leading to answers of almost all kind of algebra questions and yes faster than all other methods. The key is to really understand the terms and then use logic and sequential chain of thoughts to reach the answer.
1. It’s just input output dummy
if i ask you what is one symbol that comes to mind when you think of algebra the answer probably would be X and some other symbols but I am betting my money on X .So what is this mysterious X.
Most algebra (at-least the kind which is our focus for exams Like CAT , GMAT ) is input-output analysis or simpler still a relationship between two variables. For instance think of machine with a slot for inserting tokens and a vent which drops chocolates. for the simplest case possible think that the no of chocolates dropped by the machine is the same as the number of coins inserted .here
no of tokens inserted = Input
no of Chocolates dropped= Output
and output=input
Convention is to think of the input as X and the output as Y . So in this case
Y=X
think of this as the simplest Algebra equation possible and start from here and all other concepts can be build on this foundation .What would really help us master the topic is the Visualisation of these relations in a two-dimensional space ( a piece of paper) Where the horizontal axis represents the X axis or the input line and the Vertical axis represents the output line of Y . For our imaginary machine the input output visualisation would be as follows
The red Square represents the input and the Blue circle represents the output and the whole 2-dimensional space is divided into horizontal and vertical axis where X is the horizontal line and Y is the vertical. The blue circle represents the value of Y for a specific Value of X . Once you play the video pause it a few times and each time notice the value where the red box is placed and the value of the blue circle correspondingly.
If you touch any point on the grid and want to define it’s location with reference to the origin (the intersection of X and Y axis or the Input=0 Output=0 ) you have to do so by finding out how far away it is from the origin horizontally and vertically . For instance in the following image
This right here is the key to all future ideas , understand this and rest is just moving forward with thought and without understanding this everything else is pointless.Comparing the image with our chocolate vending machine , the image is a snapshot of inserting one coin and getting one chocolate. This information is being represented in a 2-d space where points are plotted in reference to another point called the origin.
Origin is the intersection of the X and Y axis .X axis is the input line , our task in all future analysis will be to plot a point in the grid which is a graphical representation of An Input-Output Pair.
What the output will be for a particular value of X is dictated by the function definition , in our example case the definition was provided as
Y=X
i.e the output is the same as input .
To plot the point start from the center point or the origin and then either first move left or right on the X-axis according to value of X ( right for positive , left for negative) and then move up down according to value of Y then mark the spot with a dot or a cross. Try moving your finger on the screen along the X axis first then up down. OR first move up or down according to value of Y (up for positive , down for negative ) then move left or right according to value of X .
So plotting a point is easy peasy , our maqsad is to feel the same comfort for almost all the mathematics involved .
If we for a moment consider two different input values simultaneously say X=1 and X= 2 , for each input we will get an output , Y= 2 , Y=2 , Now if we plot a point for both the set of numbers (X=1,Y=1) , (X=2, Y=2) the points would appear as
Now what if we connect the two points by drawing a line and while at it extend the line in both directions . like in the image , this line is the line represented by the relation Y=X , i.e any point on this line has X and Y values which satisfies the given relation Y=X. So a line is a series of points bound by a common relationship .
Now imagine plotting a series of points for values of X from negative to positive and the connecting the points together and drawing the line the following video should help in visualization .