# 30MSC 2023 #24: By Addition and By Subtraction

Only a few problems in Math do not require addition or subtraction at some point in their solutions. In many types, however, the sutra “By Addition and By Subtraction” play a major role. Almost all divisibility rules, for example, are based in this sutra.

Another major application is in this question taken from the Senior IVMO 2021: If 5x − y = 18 and 5y − x = 12 , what is the value of x − y ?

In solving a system of equation with two unknowns where the coefficients are interchanged, this sutra is very handy.

By Subtraction, we can easily get the answer: (5x – y = 18) – (5y – x = 12) = 6x – 6y = 6 or x – y = 1

But what if we want to know the value of x and y? We could use By Addition to get (5x – y = 18) + (5y – x = 12) = 4x + 4y = 30 or x + y = 7.5

Again By Addition, we have (x + y = 7.5) + (x – y = 1) = 2x = 8.5 and x = 4.25

By Subtraction, we get (x + y = 7.5) – (x – y = 1) = 2y = 6.5 of y = 3.25

To check, we have 5x – y = 18; 5(4.25 ) – 1(3.25) = 21.25 – 3.25 = 18

And, 5y – x = 5(3.25) – 1(4.25) = 16.25 – 4.25 – 12

Further uses of By Addition and By Subtraction are discussed in Chapter 4 of our forthcoming book, 30 Master Strategies in Computing Vol III.

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