Interpreting Solution set

All values of a variable that satisfies equality, inequality, system of equations, or system of inequalities can be termed as solution set. All the values that are in a set for whom any equation is satisfied can be interpreted with the help of any feasible solution may it be graphical or algebraic. Solution of any equation or in equation depends upon the domain of the subject and its availability.

For example if we assume a linear equation as 3x + 2y = 12, then we need to find all the values of x and y that satisfied the given equation. All the values of x and y that satisfies the above equation can be put under one definite set, such as one set would be (2,3), similar to this there can be many sets of solution available.

Let us take another example as 5x + 6y = 30. Then one of the solution set of an equation would be (6,0). Now it is true that there may be many solution-sets and when we integrate all of them together we call it as a set.

There are many different ways to solve an equation, out of which most popular is solving it by hit and trial method. The other most prominent is the matrix solution by substitution the coefficient values of the equation. Any solution set of an equation depends on the type of variable it is having.

While dealing with algebra we will come across many such equations having unique solution and having infinite solutions.

Moreover solution set examples could be found in the questions of probabilities and permutations and combinations. Algebra is that part of mathematics that deals more about numbers in the form of equations. Apart from the linear equations there more complex equations for which solution-set can be determined, more complex the equation is, and more complex is the solution-set. For linear equations there may be more than one variable, but most important is that the degree of the equation should be one.

Similarly we have quadratic equations whose degree is two, means we get two solutions for one variable each time we solve. More preciously the solution-set may consist of degenerated sets, which might be imaginary numbers, but still can provide the solution.
So hence all equations have some solutions unique, some or infinite, which together collected and can be names as solution set.

Real life problems as word problems

It is much similar to the word problems what we have seen in the mathematics. It is just about a word expression which we should have to convert it into the productive form or the algebra word problems or in algebraic expression. It is nothing but the conversion of the real word situation and translates it into the mathematical form or some expression that having a numerical type of approach. We are here to give you how to do algebra problems?

Say for example as x plus 5 means x+5, this is the conversion of the word expression into expression of mathematics. Algebra word problems help you will take from these categories more. Word problems consist of the sentences and we need to resolve into the problems very carefully and through our understanding and bit analytical skill.

As word problems are somewhat difficult so you have a good exposure of the problems before also. So practice more over these kinds of equations. Say for example as on my birthday, my weight is 45 kg. One year later I have put on x kg so which expression gives my weight correctly after one year. Sometime the equations like polynomial based problems are also there.

Now we will discuss more on algebra word problem help so that you can easily solve these kinds of problems. Say in probability you will learn about algebra 2 questions and answers type of complex problems. Firstly you have to understand what is important in the question and don’t jump to the solution, be cool and calm while solving and then form the equation after reading once then go through the needs and requirements is all there. If yes then proceed further by solving these problems with the formation of equations or out it into proper tabular format so that to avoid mistakes happened in the future.

Algebra is not about the plus and minus you should have to think how to frame the questions. Once you understand this concept it is very easier to solve the word problems. Sometimes it is long paragraph wise don’t hesitate to solve it is easier one many times, it just need only your attention and confidence to solve. Once you take your decision to solve proceed further step by step and find out the right solution. It is real life based problems we can say that because framing of the questions gives you the gist of the future or present and sometimes past also.

Rotational Symmetry

Guess an object rotated once, twice, and thrice or may be as many times, but still the observation angle and view remains same, those kinds of objects have rotational symmetry, means rotate them through their symmetrical line it will appear same.

An object might have more than one rotational-symmetry, for instance its reflections against any reflection line may also be considered as symmetrical line. The question often arrives in our mind that what is rotational symmetry. The answer is simple, any such object when rotated as many times but still appears same has rotational-symmetry.

As such talking more specifically in terms of science there are many theoretical explanations of the same, but it can be expressed as simple as stated above.

Many mathematicians have the perception that this could be the characteristics of the object, but preciously speaking this is not yet proved. Much rotational symmetry definition is under consideration, all ending with same conclusions.

Rotational symmetry of order n or n fold rotational-symmetry is known as discrete rotational form symmetry of the nth order.  For n = 1 means it is rotated 180 degrees, for n = 2 it means rotated at 120 degrees, and it goes on as the value of n changes.

We can say when a shape is rotated about any fixed point and if it comes to rest in such a position and looks exactly like the original. Some of the rotational symmetry examples are an equilateral triangle, or a tetrahedral structure, no matter how much we rotate the shape is always same, when we observe it from any angle.

Another good example is the pizza of round shape, all pieces of pizza are distributes in equal shapes and size. If the shape has rotational-symmetry then it must eighter have line symmetry or point symmetry. For example a pentagon star has five points and thus has five lines of symmetry but does not have points of symmetry.

Only an object which has point symmetry or line symmetry can have rotational form symmetry. Thus we can have lot more examples in real world. Let us take the alphabetical letters, let us talk about the letter “I”, it has two line of symmetry and one point of symmetry hence this would be considered as rotational-symmetry on the basis of its rotation. The geometry of rotational-symmetry is much better explained in rotational dynamics of physics.

What is Inverse Variation

In the given two variables if the value of one variable increases and another one decreases   that is if the values change in opposite manner then it’s said to be inverse variation.
For example if the speed increases the travelling time decreases and if speed decreases the travelling time increases

Inverse variations  occurs  when two variables such as XY are always equation to some constant K ie, one value  decreases and the other value increases

No of workers Days to complete the  work
        15               15
        10               18
         5               20

As the   number of worker decreases the days to complete the work increases.  Here inverse variations occur.

Inverse Variation Formula

Inverse-variation between two variables say  y and x is given by the formula    y  =  k/x
Here the variable K is known as the constant of proportionality.  Y varies inversely with x.
Y varies inversely with x, if there is a nonzero constant k such that xy = k or ,  y  = k/x, where k ≠ 0.
In inverse-variation its seen that , when the value of one  variable increases the value of the other value  decreases in proportion so that the product remains the same always.

Inverse Variation Word Problems

While solving inverse-variation word problems we use the formula ‘y = k/x   or any other variables relevant to the problem. Find the value of ‘k’ which is the constant of proportionality.
Plug in the value of k in the formula y=k/x .  Use the information from the given problem and solve for the variable.  An example problem is solved below.

If a car travels at the rate of 20 miles per hour it takes 1.5 hours to reach its destination.  If the same car travels at the rate of  50 miles per hour  what will be the time taken to reach its destination.

Use  the formula     t  =  k/s  where  t = 1.5 and s = 20 .  Plug in the values of t and s and find k.
         
                                   1.5  = k/20    (multiply by 20 both sides)
                                   K  = 1.5 x 20
                                   K =  30

Substitute the value of ‘k in the  equation   t = k/s
Here   k = 30  and  t  =  50
                                    t = 30/50

                                    t =  0.6 hours
The car takes 0.6 hours if it travels at the speed of 50 miles per hour.    Here it is found that as the speed of the car increases the time taken reduces.  So   there is inverse-variation here.

Dot product

There are two types of multiplication of two vectors. That is, as two types of products. One of them gives the result as a vector quantity and the other results as a scalar quantity. They are termed as cross product and dot product respectively.  The names are given so because in the first case a ‘x’ is used to indicate the multiplication and in the second case a ‘•’ indicates the same operation. Because of the fact that a dot product results in scalar quantity, the dot-product is also called as scalar dot product.

If P and Q are two vectors, then the dot-product two vectors is denoted as P • Q and its magnitude is given by lPl*lQlcos θ, where θ is the angle between the vectors P and Q. Unlike vector products, scalar products of vectors are commutative.

A typical example of a scalar product is the calculation of work done W by a force F while displacing the object by a displacement D. As per physics concept, W = lFl*lDlcos θ, where θ is the angle between the force and the direction of displacement. Although, F and D are vector quantities, the work done W is a scalar quantity. The work done by a force has no direction. .

From the definition, P • Q = lPl*lQlcos θ, let us analyze when the dot product is zero. It is obvious that it happens when P or Q or both are 0. But there is one more possibility. The scalar product becomes 0 when the angle between the vectors is 90o, even though P and Q may have finite values, since cos 90o = 0. In other words, the scalar product of two vectors acting at right angle is 0. We can illustrate this fact with a practical example. We all know that the weight W of an object placed on a smooth horizontal surface is a vector acting vertically. Suppose the object is moved horizontally by a displacement D, the vectors W and D act at right angle. Since the scalar product is 0, the work done W is 0. That is, no work is needed to be done in moving an object at a horizontal level subject to the assumption that there is no friction between the object and the horizontal surface. That is why we find that only a little amount of effort is needed to push even a heavy object horizontally. It is not zero as per theory, because the assumption of ‘no friction’ is practically impossible and the little effort only accounts for work done against the frictional force.

Mathematical Induction Definition

It is a process that is used to trueness of a given statement for all positive integers. This method includes proving the first statement true firstly from the series of infinites statement and then it is proved that the any one of the other statements from that series is also true.

The Principle of Mathematical Induction includes following steps to prove that a statement is true for all natural numbers:

1)Basis step: the statement is proved true for the least natural number which is normally 0 or 1.
2)Inductive step: In this step it is assumed that the statement is true for some natural number p that lies between least possible value of n to any natural number n, and then it is proved that the statement is true for the next natural number (p+1) also.
The value of n for basis step is decided from 0 and 1 on the basis of given question or statement.

Proof of Mathematical Induction
Let statement S(n) is false for some values of n. Let no be the least value of n such that S(no) is false. no cannot be 0, because S(0) is true. Therefore no should be 1+n1. As n1
Let us take some Examples of Mathematical Induction to solve some problems:

Mathematical Induction Proofs Examples

Q.1) Show that the sum of first natural numbers from 0 to n is given by n(n+1)/2.
Sol.) The above statement can be written mathematically as:
0+1+2…..+n = n(n+1)/2

Let us use mathematical indction process now:
Basis step: in this step we will put least possible value of n in above equation, which is 0 for this case.
S(0) =>
 0 = 0(0+1)/2
In above equation LHS is 0 and RHS also solves to 0. Hence basis step hold true and thus statement is true for n=0.

Inductive step: let us assume that the equation is true for a number p that lies between 0 and n. Now we will show that it is true for p+1 also.
S(p+1)=>
0+1+2…..+p + (p+1)= ((p+1)((p+1)+1))/2
LHS can also be written as
(p(p+1))/2 + (p+1) , as we assumed that the statement is true for p natural number.
(p(p+1))/2 + (p+1) = (p(p+1)+ 2(p+1))/2
= (p^2+p+2p+2)/2
= ((p+1)(p+2))/2  
=((p+1)((p+1)+1))/2 = RHS.
Therefore S(p+1) is true. Hence from basis and inductive steps we come to the conclusion that S(n) is true for all natural numbers.

LCM and its methods

LCM of two numbers is the smallest number (non zero) that is multiple of both.

When we add or subtract any fraction, we make use of the least common denominator and least common denominator is nothing but the least common multiple of more than two numbers.

LCM - least Common Multiplies actually the least possible common multiple of two or more than two numbers. We can find the least common multiple by using two methods: -

Methods for finding LCM of numbers:-
1. LCM Using Prime Factorization– In this method we find the prime multiples of all the numbers for which we need to find the Least common multiple.
We find the common factors among them and the uncommon factors. Least Common Multiple is product of common factors and product of uncommon factors.

LCM Examples- Suppose we need to find the least common multiple of 15 and 25.
We see that 15 = 3 times 5 and 25 is 5 times 5.
Now we see common factor is 5 and uncommon factors are 3 and 5.
So the least common multiple will be 3 times 5 times 5 which equal 75. Hence the least common multiple of 15 and 25 is 75.

2. Common division method- The other method of finding the least common multiple is the common division method in which we arrange the numbers together separated by commas.
We start dividing with the smallest prime number and go on dividing, till none of the numbers can be divided any further.

For example:- If we need to find LCM of 20 and 30 by common division method, we first divide both of them by 2, we get 10, 15. Then we divide again by 2, we get 5, 15. Now we divide by 3, we get 5, 5 and lastly by 5, we get 1, 1.
Now we multiply all the prime numbers by which we divided. They are 2, 2, 3, 5 which is 2 times 2 times 3 times 5 and that gives 60. Hence the least common multiple of 20 and 30 is 60.

We can try different problems for LCM Practice.
We can find the least common multiple of more than two numbers also.
This is usually helpful when we are adding and subtracting the fractions.