12.001 Theory of Equations

“Pure mathematics is the world’s best game.It is more absorbing than chess,more of a gamble than poker, and lasts longer than monopoly.It’s  free. It can be played anywhere – Archimedes did it in a bathtub”                                                                      –Richard j.Trudeau , Dots and Line

Solution of Linear Equations

Skipping (will be explained once taught thoroughly) the very fundamental definitions, of linear equation etc.  Let us take the very simple examples first, to know the solution,

x + y = 5                                                                     ,  2x + y = 8

Now, what here these equations represents, say x is weight of chocolate 1 and y is the weight of chocolate 2 for a child, then here we have equation which declares some conditions, x + y = 5 i.e. the sum of weight of chocolate 1 and chocolate 2 should be 5.

Here, we have many solution(infinite)  (1,4),(2,3),(3,2)…..

But if I say you have one more condition to follow,2x + y = 8,       i.e. sum of twice of weight of chocolate 1 and weight of chocolate 2 should be 8. It also may have infinite solutions (1.1, 5.8),(2,4),(3,2)…..

• A solution means values which follows both conditions simultaneously

Geometrical Representation- These two equations represents a line (What?)I mean, If I darken all point of solution in graph, I will get a line on graph.

When we have to solve them simultaneously, we would plot all the solution, of both equations on single paper and find common solution, which we say the intersection point of the two equations.

• Back substitution method

In the equation, 2x + y = 8

Here, x and y are independent, but if I impose a condition on y ( get it dependent of x) we will have a equation completely dependent of x’s value(and eventually get depended value of y)

So, x + y = 5   =>    y = 5 – x   (condition on value of y relating it by the value of x)                     2x + (5 – x) = 8    =>   5 + x = 8   =>   x = 3 and corresponding value of y = 2

• Breakdowns– The first case is easy, one solution and that too unique, but there are few condition that are not possible to get a solution that way,and for that matter by any way as they don’t have a common solution and similarly some equations have infinite solution, just see the picture, they are self explanatory.

Need of Matrix

Now, we have 3 equations, 4 equations or more variable it will get into 3D, 4D and so on, the geometry works in the same way, but it get beyond our imagination. Therefore, we build something very interesting called matrix.

• Introduction to matrixes – Matrix is represent of coefficient and numbers, say we had,

2x + y = 8 and x + y = 5 (lets represent solution using matrices)coefficient matrix, variable matrix, resultant matrix

[1  1][x ] =  [5]

[2  1][y]  = [8]

[1  1][x]  =   [1.x  1.y]  =   [5]                                .                                                                                                                                    [2  1][y]      [2.x   1.y]  =   [8]

As we say, these equations as just lines, there is another way to see the matrices, as vectors                                                                                                                                                                 x.[1] +   y[1]   =   [5]    < — i direction

[2]         [1]        [8]   < — j direction

Now, matrix can be represented as vector and since, we have 2 variables only (i.e.we have all z=0 here) All vectors on XY plane only.Now, in this case we already know the solution      I.e. x=3, y=2. So let’s take a look how does this thing work on graph.

If we see it in graph, the column matrices are vectors and x ,y are number of how many such vectors are required to get the resultant vector. Now, since we already know the solution, Let’s just draw it. If we use the property of vectors of “independent movement in space”, we can see the resultant is the sum (5,8) the position vector of required column matrix.

This is just an Introduction of  Theory of Equations, we will see each and every topic in detail, how they work, what is actually happening when we are doing some numerical step, for more refer to the video on you tube -utkarshiniedu might have better explanations.

Abhishek kumar jha

(Mathematics at Utkarshini)

for chapter notes// theory-of-equations