Learn to Solve Simultaneous Equations Mentally

Do you struggle with solving simultaneous equations?

By simultaneous I mean equations with multiple unknown variables. Generally the number of equations given will be equal to the number of equations.

Let’s take an example,
3x + 4y = 18
5x + 7y = 31

Methods Taught at Schools

In our schools, we are taught to solve for x by equating the co-efficient of y by multiplying both the equations by some constants in such a manner that you get the same resultant value and then subtracting one equation from the other.

For instance in this case, to find the value of x, we will multiply first equation by 7 and second equation by 4 and get 28 in both the cases. This is done so that we get a zero on subtracting.
3x + 4y = 18 …………………(i) x 7
5x + 7y = 31 …………………(ii) x 4

We get 2 equations, where co-efficient of y is same
21x + 28y = 126
20x + 28y = 124

Subtracting second equation obtained above from the first one we get
(21x – 20x) + (28y – 28y) = 2
Hence, x = 2

Plugging the value of x in equation (i) we get y = 3

Problems with the Above Methods

  • This method can become quite laborious, especially when the co-efficients of the unknowns are such that they have to be multiplied by large numbers to make them equal to eliminate one of the unknown by adding or subtracting as the case may be.
  • The above calculations become cumbersome, when the co-efficient(s) are large prime numbers.
  • This method involves multiple steps where we need to do multiplication and addition/subtraction.
  • Also, there is no chance of using this method to solve the problem mentally as one has to keep track of the equations and various computations.

Solving Simultaneous Equations the Smarter Way

Our new method will give us the final answer in fractions, i.e. in Numerator and Denominator for both the variables: x and y.

First, we need to find the numerator of the value of x in the above case, take the simple following steps:
Step #1: Cross-multilply the coefficient of y in the first equation by the constant term (RHS) of the second equation
Step #2: Subtract from it the cross-product of the y coefficient in the second equation and the constant term (RHS) of the first equation.

Solving Simultaneous equation

So the numerator is 4×31 – 18×7 = 124-126 = -2.

Second, we need to find the denominator of the value of x:
Step #1: Cross-multiply the coefficient of y in the first equation by the coefficient of x in the second equation
Step #2: Subtract from it the cross-product of the y coefficient in the second equation and x coefficient in the first equation.

Solving linear equations

Hence the denominator is 4×5 – 7×3 = -1
Hence, the value of x = -2/-1 = 2

Now, let’s try with a simpler example,
x+2y = 8
3x + y = 9

Using the above method, in a single line calculation you can say,
x = (2×9 – 1×8)/(2×3 – 1×1)
x = 10/5 = 2
Therefore, y = 3

Isn’t this amazingly simpler? With some practice you can comfortably apply this technique to solve simultaneous equation mentally.

If you liked this method, you must explore another Vedic Mathematics trick of solving a special class of simultaneous equations in seconds.

Do you find simultaneous linear equations difficult to solve? Do you think you can start using above method in solving equations?

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How to Select Right Math Tutor for Your Child

right math tutor for your childThe most common way to help your child improve in his or her learning is by hiring a tutor. Hiring one does not necessarily mean that a child is a slow learner. Mathematics is one of the academic subjects in which many students need help for improvement. To ensure that your child can benefit the best help, you should look for a tutor that can address the learning style and temperament of your child. This way you can expect for the best possible results.

Though most of the suggestions below are generic in nature, however, this article is written keeping in mind audience from the USA

1. Tutor’s college degree & major

When hiring a tutor, the first thing to consider is the college degree completed. If you’re hiring a tutor through an agency, you should take time verifying the degree and the major of discipline. Since you want your child to improve in mathematics, obviously it is best to hire a tutor having college major in this subject.

2. Teaching credentials and Background

As much as possible, it’s a smart idea to hire a tutor with good teaching credentials and background. It’s not enough that a tutor has the capability to impart learning and knowledge. Thus, before hiring a tutor make sure that he or she has passed the licensure examination given by the state. Keep in mind that you will entrust your child to the tutor for some period of time that’s why it’s essential to implement background check and verify the employment background. Once you know that the tutor you’ve hired is legitimate and a member of a qualified organization, you’ll have peace of mind that the teaching process will flow smoothly.

3. Rapport

The connection between the student and a tutor plays a big role to the success of the process. This makes sense of hiring a tutor who can establish a good rapport with your child particularly if there’s some behavior issues.

4. Ask your kid’s teacher

If it’s your first time to look for a tutor, you can consider recommendation from people who have been doing the same. Your child teacher at school or the school counsellor can be the appropriate individuals who can recommend a qualified tutor.

5. Check community center bulletin boards

Another way to find a qualified tutor is through community boards or library wherein tutors ate posting information.

6. Select a tutor willing to work interactively on concepts

Prior to hiring a tutor you should check first his or her capability and the concepts of teaching. Hire a tutor who is willing to utilize interactive concepts. The tutor should not only provide lectures but also interacts with the student. Likewise, as a teacher, your child’s tutor should be willing and give time in addressing the questions of your child. The tutor must be always ready to assist your child in solving math problems.

7.  Try to search a tutor who is open to feedback

Select a tutor who welcomes feedback in appositive manner. This means that before hiring one, parents should discuss with the tutor a sort of guidelines along the teaching process. He or she should also be open to discussion and suggestions from the parents in order to rule out which concept will work or doesn’t to the child.

It is very challenging to find the right math tutor to your child knowing that parents should be the first and personal teacher of a child. But, there are times when parents don’t have enough time helping a child to improve in academic subjects, thus hiring a perfect math tutor can be the last option. There’s nothing wrong hiring a tutor as long as he or she can help to improve the ability of the student and not just to earn income.

Author Bio: Andy Bell is an everyday learner and editor, working at Tutoring help. He is passionate about helping online learning businesses to achieve their goals. He loves tutoring and his favourite subjects are math and English, but he has also taken honours classes in science and history. Outside the classroom, he enjoys tennis and is an active member of the Rancho Crandon Park.

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Free Mathematics Resources Available Online

During our student life (school or post-school), search for useful knowledge resources for learning arithmetic, algebra, trigonometry or some other stream of mathematics is never ending.  Here let us discuss few such online free sites, which can be very useful in learning maths. These resources available for free on internet can help us learn fundamentals of various basic to advanced level topics of mathematics. Some of these also provide online tests for practice and solutions to thousands of questions.

Khan Academy – According to me nothing on internet can beat the zeal of Mr. Salman Khan who has produced more than 3400 videos on diverse topic and has extensively covered Mathematics from basic to advanced level in his wonderful video lectures. Best part is all the lessons are absolutely free for everyone.

www.khanacademy.org Read More

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Palindromes – These Numbers will Make You Fall in Love with Numbers

What are Palindromes?

Palindromes are very special kind of numbers. Typically a palindrome can be described as a number, word, sentence, etc. which reads same forward and backward. Specifically with regards to numbers, Palindromes are numbers which are symmetrical, i.e. they remain the same even when their digits are reversed.

For example 14641 is a Palindrome. In fact all the single digit numbers and numbers with same digit repeated are palindromes. So all numbers like 1,2,3…8,9,11,22,99,111,etc. are palindromes.

Properties of Palindrome Numbers

Property #1

Reverse a non-palindromic number and add it to the original number. We will get a palindromic number by repeating this process. We may even get a palindromic number in first go. For example, let the original number be 37 (non-palindromic). Add reverse of it 73 to 37, we get 110 (not a palindromic number). Therefore repeat the process. 110 + 011 = 121 (palindromic number). Another example, 16+61 = 77 (palindromic number).

Any number that never becomes palindromic in this way is known as Lychrel Number. The most famous Lychrel number is 196. Check out the calculations for yourself!

Property #2

A palindromic number in one base may or may not be palindromic in any other base.  For example, 1991 is palindromic in both decimal and hexadecimal (7C7)

Property #3

Certain powers of palindromes made up of digit 1,2 and at times 3 are mostly palindromes.

For example,

  • 11^2 = 121
  • 22^2 = 484
  • 101*101=10201
  • 111*111=12321
  • 121*121=14641
  • 202*202=40804
  • 212*212=44944

There are, however, an infinite number of cases as demonstrated here:

  • 11^2 = 121, 101^2 = 10201, 1001^2 = 1002001, 10001^2 = 100020001, etc.
  • 22^2 = 484, 202^2 = 40804, 2002^2 = 4008004, 20002^2 = 400080004, etc.

Property #4

All even digit palindromes are divisible by 11. There are many prime palindrome numbers also like 101, 131, 151, 181, and 191

Similar to palindromic numbers, 1089 and 6174 (Kaprekar Constant) have beautiful properties

Palindrome Challenge for You

Based on the above knowledge take this very interesting palindrome challenge –

  1. Give 2 examples of known “Lychrel Number”, other than 196.
  2. Give me the most recent palindromic date.
  3. Most of us have lived through two palindrome years, 1991 being the last one. Only 11 years separate 1991 and 2002. Most palindrome years are separated by 110 years.Has there ever been a time when two palindrome years have been separated by less than 11 years?  (Here I am not talking about single digit palindromes)

There are many other interesting features related to palindromes.  Share some special feature which you could figure out.

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Quick Multiplication up to 20 x 20

“I’m having trouble above 10×10.”

This was a statement I heard many times while interacting with students preparing for competitive examinations including CAT. This was in response to my appeal to them to memorize tables up to 20×20.

Today I am posting here on QuickerMaths.com, the method which I recommend to my students too.

How to multiply up to 20×20 in your head?

Assumption: You know your multiplication table reasonably well up to 10×10.

I am trying to explain this with an example, Read More

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What’s your favorite equation?

It has been a while since we had a survey on any mathematics topic, so today I’d like to kick one off. What’s your favorite equation, formula, identity or inequality?

Try to keep it to your top 5 so that things don’t get out of hand – but please share which equation, formula, identity or inequality amuse you the most.

So – What’s your favorite equation, formula, identity or inequality in mathematics and if possible tell us why? Over to you!

Take into consideration all the branches of mathematics

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Quick calculations for extremely large numbers

A guest post by Nandeesh H.N. of Kolkata

Quick calculations with a few logarithms

If you can remember a few logarithms, you can do many calculations quite easily without the aid of calculators or computers.

Try to remember the logarithms of just seven numbers:

Log 2 = 0.30, log 3 = 0.48, log 7 = 0.85, log 11= 1.04, log 13= 1.11, log 17 = 1.23 and log 19=1.28.

The logarithm of a composite number is equal to the sum of the logarithms of its prime factors; you can formulate the following table of logarithms:

Read More

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