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Multi Variable Equations - CMI UG Entrance 2019 - Problem 5

Number System


Multi variable linear equations are equations that have two or more unknowns (generally represented by 'x' and 'y').

Try the problem


Three positive real numbers $x, y, z$ satisfy
[
\begin{aligned}
x^{2}+y^{2} &=3^{2} \
y^{2}+y z+z^{2} &=4^{2} \
x^{2}+\sqrt{3} x z+z^{2} &=5^{2}
\end{aligned}
]
Find the value of 2 x y+

CMI UG Entrance, 2019

Multi variable Equations

6 out of 10

AOPS Intermediate Algebra

Use some hints


Assume that the given inequality is true for ( a>0 , b>0 ) and ( a+b<2 ) . Then proceed. Note that the values on the RHS of the three equations are squares of the Pythagorean triplet (3,4,5).
So draw a right-angled triangle with sides 3, 4, & 5 and then proceed.
(Say we take a triangle ABC with sides AB = 3, BC = 4, & CA = 5)

In the triangle that you were asked to construct in (Hint 1), take a point O inside it and name OA = x, OB = y and OC = z.
Now try to relate these with the equations you are provided with in the question.

Try to predict the angles AOB, BOC, & COA to make use of the cosine formula.
EXAMPLE : From the first equation and the triangle you were asked to make, it is quite obvious that angle AOB = 90 degrees.
Hence, from triangle AOB, we obtain:
\(x^2 + y^2 – 2xy cos90^\circ = 3^2\), i.e., \(x^2 + y^2 = 3^2\)

\( (a-b)^2 \geq 0 \) as \( a,b \in{R}\) .

And to get ( (1-ab)>0 ) use the well known inequality for positive reals i.e. \( AM \geq GM \) and the still unused inequality i.e ( a+b <2 ) also .

\( a>0 , b>0 \Rightarrow \sqrt{ab}>0 \Rightarrow( 1+ \sqrt{ab})>0  \)  \(a>0 , b>0 , a+b <2 \Rightarrow  1 > \frac{a+b}{2} \geq \sqrt {ab} \ \Rightarrow 1 > \sqrt{ab} \ \Rightarrow ( 1 - \sqrt{ab}) >0 \ \Rightarrow (1 - \sqrt{ab}) (1+ \sqrt{ab}) >0 \ \Rightarrow (1 - ab)>0 \)

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