An algebra problem by Yan Yau Cheng

Algebra Level 3

Let x x , y y , and z z be reals satisfying the following:

z = 2 x + 4 y + 5 z = 2x + 4y + 5 − x 2 + x y + y 2 = 2 z 2 + 8 -x^2 + xy + y^2 = 2z^2+8 5 x 2 + 3 x y = − 8 − z 2 5x^2 + 3xy = -8-z^2

If y = − a b y = -\frac ab for positive co-prime integers a a and b b , find a + b a+b


The answer is 8.

Yan Yau Cheng by Yan Yau Cheng , 20, Hong Kong

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

By combining Equation (2) and (3), we get: \­[ 4{ x }^{ 2 }+4xy+{ y }^{ 2 }={ z }^{ 2 }\ This\quad implies\quad that\quad { (2x+y) }^{ 2 }={ z }^{ 2 }\ Therefore\quad z=2x+y;\ 2x+y=2x+4y+5\ y=\frac { -5 }{ 3 } . \­]

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...