Solve the simultaneous equations: x^2 + 8x + y^2; x - y = 10.

Label the two equations.

x² + 8x + y² = 84 (1)
x - y = 10 (2)

Rearrange (2) to get y = x - 10 and substitute for y in (1) to get x² + 8x + (x - 10)² = 84. Expanding and collecting like terms gives 2x² -12x + 16 = 0 (3). Dividing (3) through by 2 gives x² - 6x + 8 = 0 (4). Factorising (4) gives (x - 2)(x - 4) = 0 so either x = 2 and y = -8 or x = 4 and y = -6.