Cambridge O Level Additional Mathematics · Syllabus 4037 · Simultaneous Equations
Ordered Pair
What is Ordered Pair?
A solution of a system of two equations in two unknowns, written (x, y), in which the order of the two components carries meaning: the first is the value of x and the second the value of y that arose from it. The pairing is determined by the equation used to recover the second unknown, so the components cannot be recombined with those of another solution, and a set of x values listed separately from a set of y values is not an answer until the correct pairings are stated.
This definition is part of the Simultaneous Equations chapter in Cambridge O Level Additional Mathematics.
Ordered Pair in context
A simultaneous solution is one ordered pair \((x,y)\) that satisfies both original equations at the same instant. In Additional Mathematics at least one of those equations is usually not a straight line, so the reduced equation is a quadratic and there is normally more than one ordered pair. Your job is to find every pair, keep each \(x\) welded to its own \(y\), and prove each pair works in both originals.
Common mistakes with Ordered Pair
- “My answer is \(x=2,5\) and \(y=3,0\).” WHY IT FAILS Two coordinate lists do not state which value goes with which. The reader has to guess your matching, and an answer that has to be guessed at has not been stated — even when every number in it is correct. FIX Report ordered pairs: \((2,3)\) and \((5,0)\). Where the question says “coordinates”, use coordinate notation.
- “\((x-y)^2=4\), so \(x-y=2\).” WHY IT FAILS Every positive number has two square roots. Taking only the positive branch halves the solution set, and in a symmetric system that means losing two of the four ordered pairs. FIX Write \(\pm\) at the moment you take the root, not afterwards: \(x-y=\pm2\).
Questions students ask about Ordered Pair
What does it mean to solve simultaneous equations?
It means finding every ordered pair \((x,y)\) that satisfies both original equations at the same time; graphically each pair is a point where the two graphs intersect. In Additional Mathematics at least one equation is usually not linear, so the reduced equation is a quadratic and there are normally two pairs, though there may be one (a repeated root) or none (a negative discriminant). Two numbers are not an answer; a pair is an answer.
How should the final answer to a simultaneous equations question be written?
As a complete set of correctly matched ordered pairs, for example \((3,1)\) and \((-1,-3)\), with surds left exact. Writing \(x=3\) or \(x=-1\) is half a solution, and listing \(x=2,5\) and \(y=3,0\) separately forces the reader to guess the matching. Each \(y\) was generated by its own \(x\), so cross-pairing produces a point such as \((3,-3)\) that lies on neither graph. Give the number of pairs the discriminant predicts.