MSC #28: By Addition and By Subtraction: Solving a system of equations:
“The equations of two lines are x + y = 14, and x − y = 2 . What are the coordinates of the point of intersection? A (4, 10); B (10, 4); C (7, 7); D (6, 8); E (8,6). “
This question, given in the intermediate level of the 1st International Vedic Mathematics Olympiad, is a variation of the common problem given in elementary school math competitions: “The sum of two numbers is A while their difference is B. Find the two numbers.”
This is easily solved using the Sutra “By Addition and By Subtraction”, which is the technique discussed in the 28th of our 30 MATH-Inic Specials for Christmas series.
The use of the Sutra ”By Addition and By Subtraction” is very important in solving simultaneous linear equations. In the IVMO 2021 question, we have two equations, x + y = 14 and x – y = 2. Adding the two equations will eliminate y, resulting in 2x = 16 or x = 8, while subtracting the second from the first will give 2y = 12 or y = 6. An interesting application of this sutra is where the coefficients of the x and y variables are interchanged as can be seen in our example for today which was taken from our book Algebra Made Easy as Arithmetic, p. 53. Addition and subtraction of the two equations will result to having the coefficients of the x and x terms numerically equal. This will enable us to reduce both the sum and difference to (x + y) and (x-y) forms. Then we will apply “by addition and by subtraction again: I. 8x – 3y = 22 II. 3x – 8y = – 33 III. 11x – 11y = – 11; by addition I + II; IV. x – y = – 1; simplify equation III V. 5x + 5y = 55; by Subtraction I – II VI. x + y = 11; simplify equation V VII. 2x = 10; x = 5; by addition IV + VI VIII. 2y = 12; y = 6; by subtraction VI – IV
