This System Looks Impossible. One Symmetry Cracks It.

Two numbers. Square the first and subtract the second, you get 133. Square the second and subtract the first, you also get 133. And the two numbers are not equal. Find them. a^2 - b = 133 b^2 - a = 133, with a not equal to b It looks like it needs brute force. It does not. The whole puzzle turns on one idea: symmetry. Swap a and b and the first equation becomes the second, so the system does not care which name goes with which number. That single fact does all the work. Subtract the two equations and the right side cancels to zero. The left side is a difference of squares plus a leftover, and both pieces share the factor a minus b. Factor it out: (a - b)(a + b + 1) = 0. Since a is not equal to b, the first factor is out, so a + b = -1. Now add the two equations: a squared plus b squared, minus a plus b, equals 266, and since a + b = -1 that gives a squared plus b squared = 265. The identity (a + b) squared = a squared plus b squared plus 2ab then pins the product: ab = -132. The twin identity (a - b) squared = a squared plus b squared minus 2ab gives 529, so a - b is plus or minus 23. With the sum -1 and the difference plus or minus 23, two clean linear systems give (a, b) = (11, -12) and (-12, 11). Two answers that are the same pair swapped. That mirror is the symmetry from the very first step coming back around. The symmetry did not just help solve it. The symmetry was the answer. This is a simultaneous equations puzzle, but not the kind you brute force. No quadratic formula, no guessing, just symmetry, a difference of squares, and the sum and product identities. Chapters: 00:00 Two squares, one number 00:22 Spot the symmetry 00:41 Subtract the equations 01:04 Factor the difference of squares 01:31 Reject a = b, keep a + b = -1 01:58 Add the equations 02:24 a squared plus b squared = 265 02:45 The product ab = -132 03:19 The difference a - b = 23 or -23 03:53 Solve the two cases 04:24 The symmetry was the answer Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com Stream the music on Spotify: https://open.spotify.com/artist/2SRhE... #math #algebra #simultaneousequations Math, Arithmetic, Basic Math, Math Tutorial, Solve Math Problems, Order of Operations, Math Challenge, Step-by-Step Math, Calculation Tips, Math Brain Teaser, Quick Math, Mathematical Expressions, Math Tricks, Problem Solving, Easy Math, Math for Beginners, Mental Math, Number Operations, Math Puzzles, Learn Math, Educational Math, Math Lesson, Math Skills, Math Practice, Math Explained, Solve Equations, Simple Calculations, Math Video, maths, PEMDAS, BODMAS, BIDMAS, GEMDAS, order of operations explained, simplify expressions, evaluate expressions, parentheses exponents, multiplication and division, addition and subtraction