Solve Math Problems Better Under the Guidance of Expert Tutor

Learn simple steps and techniques for solving complex Math problems from highly experienced tutors. Getting help from qualified tutors will improve your Math problem-solving skills and also make you confident during examinations.

Math has a high importance as it is widely used in several fields like Banking, Finance, Engineering, Research, Science and Technology and many more. By utilizing mathematical skills, students can improve their problem solving capabilities. Needless to say, mathematical concepts are required
in every walk of life and it is best  to learn the subject in a detailed manner. Learning this subject from a well-qualified tutor will certainly upgrade students knowledge and make them better in dealing with tricky and difficult Math sums. Apart from this, Math also helps students to understand the world in a better way as mathematical principles are being used in day to day life of every individual.

It is good to learn Math from an experienced tutor who possess sound knowledge of the subject. Whether you opt for a live tutor or an online tutor, it is important to understand Math concept in a step-by-step manner. A live tutor will teach you the subject in a classroom or in a learning center but online tutor will give you the freedom to choose a preferred location and a convenient time before starting a session. Learning problem-solving techniques under the supervision of an Online Math tutor will be quite advantageous as it will makes you confident, save your time and cost.

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Linear Programming Simplex Method

In Linear Programming Simplex Method is a method used for problem solving. This method is used for a problem of the form:
Minimize or maximize : c.x
Subject to
Ax=b, xi ≥0  
X is the variables of the problem (x1,x2…) while c is coefficients of the objective function (c1,c2,…). A is a matrix and b are constants≥0.
To solve the linear programming problems we must follow standard form which is achieved as:
A new variable should be added for all those variables whose upper bound is other than zero such that the new variable is the difference between original variable and upper bound.
For example: if x≥3 then introduce a new variable y such that
Y=x-3 and x=y+3.
For rest of the inequality constraints, new variable known as slack variable is introduced to remove inequalities. Like:
X+2y≤3 and –x+3y≥2 is replaced as:
X+2y+z=3 and –x+3y-z=2 such that z≥0.
Let us solve some problems using The Simplex Method

Question) Use Simplex Method to solve the following:
P=3x+4y subject to:
x+y≤4
2x+y≤5
x≥0,y≥0

Solution)  Since we have two constraints, we will introduce 2 slack variables p and q:
x+y+p=4
2x+y+q=5
We rewrite our objective function as −3x−4y+P=0 and thus system of equations become:
x+y+p=4
2x+y+q=5
−3x−4y+P=0
This gives initial simplex table:
X y p q P
1 1 1 0 0 4
2 1 0 1 0 5
-3 -4 0 0 1 0

Find the column with the most negative entry among x,y,p,q and P (here this is −4). Find pivot row by dividing each entry in the constant column by the entry in the corresponding in the pivot column. In this case, we get 4/1 as the ratio for the 1st row and 5/1 for the ratio in the 2nd row. The pivot row is the row corresponding to the smallest ratio which is 4 in this case. So our pivot element is in the 2nd column, 1st row =1.Now, perform the following row operations to convert the pivot column to a unit column:
R2→R2−R1
R3→R3+4R1
So, simplex table is changed to:
X Y p q P
1 1 1 0 0 4
1 0 -1 1 0 1
1 0 4 0 1 16
The variables are given the value in the constant column in the row where a value 1 is in the unit column. All variables above a non-unit column is given 0 value. So y=4, p=1, P=16, x=0, and q=0.
Thus, maximum occurs when x=0, y=4 and the maximum value is 16.
This is how Simplex Method Solver works.

Change of base formula

In logarithms, many times we encounter with the problem when there is a logarithm whose value is not known to us for that particular base, but we know its value for some other base.
For such cases we work by changing the base of logarithms by using Logarithmic Change of Base Formula which is easy to implement.

The Change of Base Formula for logarithm:
log a (x) = log b (x) / log b (a)
Here ‘a’ is base of logarithm of x before change and, b is base of logarithm of x after changing the base.
So above Log Change Base Formula can be used when you need to change base of logarithm from ‘a’ to b provided that the new base is useful to you, otherwise changing a base will not be useful.
Let us see proof of Logarithms Change of Base Formula and then we will go through some examples:

Proof:
We know that Raising ‘a’ (base of log a (x)) with the power of log a (x) gives x:
 x = a^(〖log〗_a  (x))………………………………………….(1)
Similarly, raising b with the power of log b (a) gives a:
a = b^(〖log〗_b  (a))        …………………………………………(2)
When we replace a in (1) with b^(〖log〗_b  (a)) given in equation (2), we get:
x = a^(〖log〗_(a ) (x)) =〖〖(b〗^(〖log〗_b  (a)))〗^(〖log〗_a  (x)) =  b^(〖log〗_b  (a)  〖* log〗_(a ) (x))     ………(3)
Applying log b() on both sides of equation (3) we get:
log b (x) = log b 〖(b〗^(〖log〗_b  (a)  〖* log〗_(a ) (x)))    
Using the log power rule we get:
log b (x) = log b (a)*log a(x)
Log a (x) = log b (x) / log b (a)
Now let us see some examples using the above formula:

Example 1) Change the base of following logarithm to 2: log12 64.
Solution) Using change of base formula this can be evaluated as:
log12 (64) = log2 (64)/log2 (12)

Example 2) Solve the following by changing base to e: log10ee
Solution) log10 ee = loge ee / loge 10
= e/ln 10 (loge 10 is written as ln 10)
= e/2.302   (as ln 10 =2.302; use calculator for this)

Example 3) Solve log9 27.
Solution) here it is difficult to find the value directly so we will change the base to 3 as 9 and 27 is square and cube numbers of 3 respectively.
Log 9 27 = log 3 27 / log3 9
= log3 33 / log3 32
= 3/2   (As loga bc = c(loga b) and also loga a = 1; these are other properties of logarithms)

Elementary row operations

A matrix is an arrangement of expressions, defined in general terms. The items that are arranged are called as elements. We repeat that it is only an arrangement; thereby a matrix does not suggest any algebraic operation between the elements. Due to this fact a matrix can undergo certain operations with its rows, called as matrix elementary row operations or simply as row operations. The row operations are also called as row transformations. Such transformations can be done on columns also.  Let us study the elementary row operations one by one.
1) In a matrix, a row or a column can be interchanged. For example,

a   b   c     can be interchanged as, d e f  or,  as,  a c b                                                                                                              d   e   f                                   

a   b   c             

d   f   e                                                                                                                                                                                 g   h   i

i    g   h            

g   i    h 

In the above example, the row interchange isdenoted as R2 <->R1 and the column interchange is denoted as C2 <-> C3.

2) A row or column can be modified by multiplying by a non- zero real number.For example,

a   b   c     can be modified as,   k[a]  k[b]   k[c]    or,  as,  k[a b c]                                                                                                             d   e   f                               d       e     f            k[d  e   f]                                                                                                                                                                                g   h   i                               g        h     i            k[g  h  I ]

where, ‘k’ is a non-zero real number. These transformations are respectively denoted as R1 -> kR1 and as C1 -> kC1

3) A row or column can be modified by multiplying by a non- zero real number. For example,

a   b   c     can be modified as,   a + kd    b + ke   c + kf      or,  as,   a + kb   b   c                                                                                                              d   e   f                                            d            e          f                        d + ke    e   f                                                                                                                                                                                 g   h   i                                            g             h          i                        g + kh    h   i

where, ‘k’ is a non-zero real number. These transformations are respectively denoted as R1 -> R1 + kR2 and as C1 -> C1 + kC2

These elementary transformations are extremely useful in further topics of matrices, like finding inverse of a matrix.For example, if A is an invertible matrix and B is its inverse, the formulas are,

A = I A and I = BA where I is the identity matrix of the same order.

Start with the equation A = IA.

Plug in the given matrix for A only on the left side and write only the identity matrix times A on the right. That is let the symbol A or the right remain as symbol A.

By repeated elementary row transformations, try to reach in the equation form I = BA. Then the matrix represented by B is the inverse of A.

Understanding of end behavior of functions

End behavior definition:
With respect to polynomial functions, the end behavior of the function refers to the direction and position of the two ends of the function. For example the graph of the function y = x^2, would be as follows:

Note that both the ends of the graph are going upwards. However if we see the graph of the function y = x^3, it would look as follows:

In the above graph, both the ends of the graph point in different directions.

End behavior of polynomial functions:
Polynomial functions have primarily 4 types of end-behaviors. They are as follows:
(a) Both ends going up.

(b) Both ends going down.

(c) Left end going up and right end going down.

(d) Right end going up and left end going down.

It is possible to know the end behavior of a polynomial function without actually graphing it. We use the end behavior model for that. The following are the end behavior rules that help us model the end behavior of polynomial function.

(1) If the degree of the polynomial function is even, then both the ends would point the same direction. Therefore for an even degree polynomial function, both ends would either go up or both the ends would go down.

(2) If the degree of the polynomial function is odd, then both the ends would point in opposite direction. Therefore for an odd degree polynomial function there are two possibilities, either left ends goes up and right end goes down or left end goes down and right end goes up.

(3) If the coefficient of the leading term is positive, then the right end of the polynomial function would always point up. Therefore if the polynomial is of even degree with positive leading coefficient, then both ends would point up, and if the polynomial is odd with positive leading coefficient, then as right end points upward, the left end would point down wards.

(4) If the coefficient of the leading term is negative, then the right end of the polynomial function would always go down. Therefore if it is an even degree polynomial with negative leading coefficient, then both the ends would go down whereas, if it is an odd degree polynomial with negative leading coefficient, then as right end goes down, left end would go up.

End behavior asymptotes:
Polynomial functions do not have asymptotes. However, if it is a rational function, then it would have asymptotes.