What is $x^2+y^2$ ?

Algebra Level 3

$\begin{cases} x^2 = 8x + y \\ y^2 = x + 8y\end{cases}$ If the above equation hold for real $x,y$ provided $x \neq y$ , then determine the numerical value of $x^2 + y^2$ .

The answer is 63.

1 solution

JohnDonnie Celestre
May 3, 2014

$x^2 - y^2 = 7x - 7y$ $x^2 - y^2 = 7 (x-y)$ The assumption $x \neq y$ implies, $x + y = 7$

$x^2 + y^2 = 9x + 9y$ $x^2 + y^2 = 9(x + y)$ $x^2 + y^2 = 9(7) = \boxed{63}$

How did you get $x+y=7$ ?

Micah Wood - 7 years ago

From $x^{2} - y^{2} = (x-y)(x+y)$ . Because $x-y \neq 0$ so we can divide them by $x-y$ .

Samuraiwarm Tsunayoshi - 6 years, 12 months ago

$x^{2}=8x+y...eq(1)$ $y^{2}=x+8y...eq(2)$ Substrate eq(1)by eq(2) $x^{2}-y^{2}=7x-7y$ $(x+y)(x-y)=7(x-y)$ $\boxed{x+y=7}$ $x^{2}+y^{2}=9x+9y$ $x^{2}+y^{2}=9(x+y)$ $x^{2}+y^{2}=9×7$ $x^{2}+y^{2}=63$

Chinmoyranjan Giri - 7 years ago

the same way i did.

roland casuga - 7 years ago

$x^2+y^2=9(x+7)$

$x^2-y^2=7(x-y)$

Multiply $x^2+y^2$ by $x^2-y^2$ ,

$(x^2+y^2)(x^2-y^2)=63(x^2-y^2)\\ \Rightarrow x^2+y^2=63$

Jung Hyun Ran - 6 years, 11 months ago

can you explain how you got in your brain "instead of focusing on $x^2+y^2$ i should go for $x^2-y^2$" ??

super nigfga - 7 years ago

If you focus on $x^{2} + y^{2}$ , you will need to find the value of $x+y$ . Unless you try something new, you'll stuck there forever. Dun be afraid of breaking teh rulezzz!!!!11oneoneone

Samuraiwarm Tsunayoshi - 6 years, 12 months ago

It's simple algebra a^2 - b^2 = (a+b)(a-b)

Manu Kamath - 7 years ago

Hahaha I used brute force and a quartic :') nice to know that there's an easier way

Morgan Colbeck - 6 years, 10 months ago

ang galing.. GG nung solution.

Justin Lacsamana - 6 years, 10 months ago

What is x 2 + y 2 x^2+y^2 x 2 + y 2 ?

The answer is 63.

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What is $x^2+y^2$ ?