Solve the following system of simultaneous linear equations by the substitution method:

mx - ny = m² + n²

x + y = 2m

Given,

mx - ny = m² + n² ……(i)

x + y = 2m …….(ii)

From eqn. (ii) we get,

x = 2m - y …….(iii)

Substituting value of x from eqn. (iii) in eqn. (i) we get,

⟹ m(2m - y) - ny = m² + n²

⟹ 2m² - my - ny = m² + n²

⟹ 2m² - m² - n² = my + ny

⟹ m² - n² = y(m + n)

⟹ y = $\dfrac{\text{m}^2 - \text{n}^2}{\text{m} + \text{n}} = \dfrac{(\text{m} - \text{n})(\text{m} + \text{n})}{\text{m} + \text{n}}$ = (m - n).

Substituting value of y in eqn. (iii) we get,

x = 2m - y = 2m - (m - n) = 2m - m + n = m + n.

Hence, x = m + n and y = m - n.