A quintic equation?! Think differently

Algebra Level 2

$\large {\begin{cases} {x-y=7} \\ {x^2+y^2 = 85} \end{cases} }$

Let $x$ and $y$ be positive real numbers that satisfy the system of equations above. What is the value of $x^5 - y^5$ ?

2 solutions

Denton Young
Jun 29, 2015

Since x - y = 7, x = y + 7

So: $(y + 7)^{2}$ + $y^{2}$ = 85

Multiplying out, simplifying and factoring, (y-2)(2y + 18) = 0

So y = 2 or y = -9. But x and y are both positive. So y = 2, x = 2 + 7 = 9, and $9^{5}$ - $2^{5}$ = 59049 - 32 = 59017

Prakhar Gupta
Jul 5, 2015

x-y=7 and x^2+y^2=85 (x-y)^2=x^2+y^2-2xy 7×7=85-2xy xy=18

(x+y)^2 =x^2 + y^2 +2xy x+y=11 Eq1 ----x-y=7 Eq2------ x+y=11 X=9 and y= 2 9^5 -2^5=59017