SOLUTION TO BRAINTEASER NO. 5 DISCLAIMER This solution is but one of many possible methods for solving this problem. Various other solutions may lead to the same answers. This specific solution was provided by a member of the University of the Philippines Civil Engineering Society.
PROBLEM:
How many real solutions does this system of equations have? x2 – y2 = z y2 – z2 = x z2 – x2 = y
{
SOLUTION: Notice that the three equations define three identical hyperbolic paraboloids with each one simply oriented along its own axis. Plotting the equations as contours will show you that the system intersects at a finite number of points. Analytically speaking, setting certain values for either x, y, or z would generally create an overdetermined and inconsistent system of three equations with two unknowns. However, for certain values of either of the three variables, the overdetermined system becomes consistent; therefore inferring that the number of solutions of this system is finite. To determine the number of solutions, we opted to solve the problem with illustration. For visualization, here’s the graph of the system bound in 1x1x1 and 10x10x10 3D space:
UNIVERSITY OF THE PHILIPPINES CIVIL ENGINEERING SOCIETY Department of Civil Engineering College of Engineering and Agro-industrial Technology University of the Philippines Los Baños
Consider the contours formed by the three equations on a plane z = 0: 𝑥2 − 𝑦2 = 0 This would yield the following system: { 𝑦 2 = 𝑥 −𝑥 2 = 𝑦 Solve for x: 𝑥2 − 𝑦2 = 0 ⇒ 𝑦2 = 𝑥2 𝑦2 = 𝑥 ⇒ 𝑥2 = 𝑥 x2 – x = 0 x (x – 1) = 0 x=0 | x=1 Use computed values of x to solve for y: −𝑥 2 = 𝑦 – (0) = y – (1) = y y=0 | y = –1 You now obtained two solutions: (0,0,0) and (1,-1,0). Graphically, the system would be as such: 𝑥2 − 𝑦2 = 0
𝑦2 = 𝑥
−𝑥 2 = 𝑦
UNIVERSITY OF THE PHILIPPINES CIVIL ENGINEERING SOCIETY Department of Civil Engineering College of Engineering and Agro-industrial Technology University of the Philippines Los Baños
Apply the same procedure on planes x = 0 and y = 0 and you’d arrive at the following graphs and solutions, respectively:
z
z
y
Solutions: (0,0,0) and (0,1,-1)
x
Solutions: (0,0,0) and (-1,0,1)
Notice that the graphs are identical, simply rotated. This means that whatever happens in plane z = n also happens in planes x = n and y = n, only rotated, where n is any real number. So let us see what happens beyond z = 0. At z = 1:
At z = 2:
UNIVERSITY OF THE PHILIPPINES CIVIL ENGINEERING SOCIETY Department of Civil Engineering College of Engineering and Agro-industrial Technology University of the Philippines Los Baños
At z = 3:
At z = 10:
Basically, all that happens is that all three graphs elongate yet no longer intersect at a singular point. Therefore, the solutions that we’ve found are the only real solutions to the problem: (0,0,0), (1,-1,0), (0,1,-1), and (0,0,0).
ANSWER:
4
UNIVERSITY OF THE PHILIPPINES CIVIL ENGINEERING SOCIETY Department of Civil Engineering College of Engineering and Agro-industrial Technology University of the Philippines Los Baños
