X^2-Y^2 =40......(i) and X+Y=10, solve this simultaneously please?

x^2 - y^2 = 40
factor
(x - y)(x+y) = 40
x+y = 10
substitute

10(x-y) = 40
x-y = 4

now you have two linear equations.

x+y = 10

x = 7
y = 3

x^2 - (10 - x)^2 = 40

x^2 - (100 - 20x + x^2) = 40
x^2 - 100 + 20x - x^2 = 40

-100 + 20x = 40

I assume you can solve that linear equation.

x=10-y
(10-y)^2-y^2=40
100-20y+y^2-y^2-40=0
-20y+60 =0 devide by 2
-10y=-30 then y=-30/-10 =3
x=10-3=7

X = 7, Y = 3

x+y = 10
y=10-x

x^2-y^2 = 40
x^2 - (10-x)^2 = 40
x^2 -(100-20x+x^2) = 40
x^2 -100 +20x -x^2 = 40
20x-100 = 40
20x =140
x= 7
y=10-x=10-7 = 3

x=7
y=3