A Nice Math Olympiad System of Equations - Simplified!

In this math algebra video, we will solve a mathematics Olympiad system of equations.

[chemnobeliumlab]

solved like cutting a MOLTEN butter!!!, but your solution (changing variable) was so pretty :)

[nepheo]

Another solution would be below.

## Given
EQN1: y - x = 96
EQN2: sqrt(x) + sqrt(y) = 16
## Derive x with EQN2
sqrt(x) = 16 - sqrt(y)
x = (16 - sqrt(y)^2
## Expand with (a-b)^2 = a^2 - 2ab + b^2.
x = 256 - 32sqrt(y) + y
## Substitute x to EQN1
y - 256 + 32sqrt(y) - y = 96
32sqrt(y) - 256 = 96
32sqrt(y) = 352
sqrt(y) = 11
## Substitute sqrt(y) to EQN2
sqrt(x) + 11 = 16
sqrt(x) = 5
## Obtain solutions by squaring on both sides
y = 121
x = 25

[walterwen]

A Nice Math Olympiad System of Equations: y – x = 96, √y + √x = 16; x, y = ?
y > 96 > x > 0, y – x = (√y + √x)(√y – √x) = 96
√y – √x = 96/(√y + √x) = 96/16 = 6; √y – √x = 6, √y + √x = 16
2√y = 16 + 6 = 22, √y = 11, y = 121; 2√x = 16 – 6 = 10, √x = 5, x = 25
Answer check:
y – x = 121 – 25 = 96; Confirmed, √y + √x = 11 + 5 = 16; Confirmed
Final answer:
x = 25, y = 121