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By Addition and By Subtraction

As I was browsing through the questions in yesterday’s 1st International Vedic Mathematics Olympiad (IVMO 2021), I saw that two questions which appeared in both the Intermediate and Senior Olympiads were exactly among the topics of my online synchronous meeting with my AMEA 2021-D class this morning.

The techniques discussed in  Chapter 18 – By Addition and By Subtraction and Chapter 19 – Solving Simultaneous Equations : x and y coefficients interchanged of my Algebra Made Easy as Arithmetic book can be used to solve mentally each of the two Olympiad questions in less that 5 seconds.

The first question was: The equations of two lines are x + y = 14 and x – y = 2. What are the coordinates of their point of intersection?

This can be solved mentally using the Vedic Math Sutra By Addition and By Subtraction. By adding the two equations we can eliminate the y variable giving 2x = 16 or x = 8.

By subtracting the second equation from the first, we can eliminate the x variable and obtain 2y = 12 or y = 6. Therefore, the coordinates of the point of intersection, (x, y) is (8, 6).

Even elementary school learners can solve this problem mentally when expressed in this form: The sum of two numbers is 14, while their difference is 2. Find the numbers.

The second question seems harder to solve: If 5x – y = 18 and 5y – x = 12, find the value of (x – y).

A closer look at the problem will show that the coefficients of x and y are interchanged in the two equations. We can simply subtract the second equation from the first to 6x – 6y = 6 or (x -y) = 1.     

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