4.1: Simultaneous linear equations and symmetry

Problem 92 Dad took our new baby to the clinic to be weighed. But the baby would not stay still and caused the needle on the scales to wobble. So Dad held the baby still and stood on the scales, while nurse read off their combined weight: 78kg. Then nurse held the baby, while Dad read off their combined weight: 69kg. Finally Dad held the nurse, while the baby read off their combined weight: 137kg. How heavy was the baby?

The situation described in Problem 92 is representative of a whole class of problems, where the given information incorporates a certain symmetry, which the solver would be wise to respect. Hence one should hesitate before applying systematic brute force (as when using the information from one weighing to substitute for one of the three unknown weights – a move which effectively reduces the number of unknowns, but which fails to respect the symmetry in the data).

A similar situation arises in certain puzzles like the following.

Problem 93 Numbers are assigned (secretly) to the vertices of a polygon. Each edge of the polygon is then labelled with the sum of the numbers at its two end vertices.

(a) If the polygon is a triangle ABC, and the labels on the three sides are c (on AB), b (on AC), and a (on BC), what were the numbers written at each of the three vertices?

(b) If the polygon is a quadrilateral ABCD, and the labels on the four sides are w (on AB), x (on BC), y (on CD), and z (on DA), what numbers were written at each of the four vertices?

(c) the polygon is a pentagon ABCDE, and the labels on the five sides are d (on AB), e (on BC), a (on CD), b (on DE), and c (on EA), what numbers were written at each of the five vertices?

In case any reader is inclined to dismiss such problems as “artificial puzzles”, it may help to recall two familiar instances (Problems 94 and 96) which give rise to precisely the above situation.

Problem 94 In the triangle ABC with sides of lengths a (opposite A), b (opposite B), and c (opposite C), we want to locate the three points where the incircle touches the three sides - at point P (on BC), Q (on CA), and R (on AB). To this end, let the two tangents to the incircle from A (namely AQ and AR) have length x, the two tangents from B (namely BP and BR) have length y, and the two tangents from C (namely CP and CQ) have length z. Find the values of x, y, z in terms of a, b, c.

The second instance requires us first to review the basic properties of midpoints in terms of vectors.

Problem 95

(a) Write down the coordinates of the midpoint M of the line segment joining Y = (a, b) and Z = (c, d). Justify your answer.

(b) Position a general triangle XYZ so that the vertex X lies at the origin (0,0). Suppose that Y then has coordinates (a, b) and Z has coordinates (c, d). Let M be the midpoint of XY, and N be the midpoint of XZ. Prove the Midpoint Theorem, namely that

“MN is parallel to YZ and half its length”.

(c) Given any quadrilateral ABCD, let P be the midpoint of AB, let Q be the midpoint of BC, let R be the midpoint of CD, and let S be the midpoint of DA. Prove that PQRS is always a parallelogram.

Problem 96

(a) Suppose you know the position vectors p, q, r corresponding to the midpoints of the three sides of a triangle. Can you reconstruct the vectors x, y, z corresponding to the three vertices?

(b) Suppose you know the vectors p, q, r, s corresponding to the midpoints of the four sides of a quadrilateral. Can you reconstruct the vectors w, x, y, z corresponding to the four vertices?

(c) Suppose you know the vectors p, q, r, s, t corresponding to the midpoints of the five sides of a pentagon. Can you reconstruct the vectors v, w, x, y, z corresponding to the five vertices?

The previous five problems explore a common structural theme - namely the link between certain sums (or averages) and the original, possibly unknown, data. However this algebraic link was in every case embedded in some practical, or geometrical, context. The next few problems have been stripped of any context, leaving us free to focus on the underlying structure in a purely algebraic, or arithmetical, spirit.

Problem 97 Solve the following systems of simultaneous equations.

(a) (i) x + y = 1, y + z = 2, x + z = 3

(ii) uv = 2, vw = 4, uw = 8

(b) (i) x + y = 2, y + z = 3, x + z = 4

(ii) uv = 6, vw = 10, uw = 15

(iii) uv = 6, vw = 10, uw = 30

(iv) uv = 4, vw = 8, uw = 16

Problem 98 Use what you know about solving two simultaneous linear equations in two unknowns to construct the general positive solution to the system of equations:

$$u^a{v}^b=m,u^c{v}^d=n.$$

Interpret your result in the language of Cramer’s Rule. (Gabriel Cramer (1704–1752)).

Problem 99

(a) For which values b, c does the following system of equations have a unique solution?

$$x+y+z=3,xy+yz+zx=b,x^2+y^2+z^2=c$$

(b) For which values a, b, c does the following system of equations have a unique solution?

$$x+y+z=a,xy+yz+zx=b,x^2+y^2+z^2=c$$