Question

Please solve this with all steps. Topic simultaneous equation. ​

Answer 1

Solution:-

We have

$\rm \: : \implies \: mx - ny \: = {m}^{2} + {n}^{2} \: \: \: \: \: \: \: .....(i)$

$: \implies \: \rm \: x \: - y \: = 2n \: \: \: \: \: \: \: \: \: ......(ii)$

Using substitution method

Take ii eq , we get

$: \implies \: \rm \: x \: - y \: = 2n \: \: \: \: \: \: \: \: \:$

$\rm : \implies \: x = 2n + y \: \: \: \: \: \: \: \: \: \: \: ...(iii)$

Now substitute iii eq on i eq , we get

$\rm \: : \implies \: mx - ny \: = {m}^{2} + {n}^{2} \: \: \: \: \: \: \:$

$\rm \: : \implies \: m(2n + y) - ny \: = {m}^{2} + {n}^{2} \: \: \: \: \: \: \: .....(i)$

$\rm : \implies2nm \: + my - ny = {m}^{2} + {n}^{2}$

$: \implies \rm \: y(m - n) = {m}^{2} + {n}^{2} - 2mn$

We can write m² + n² - 2mn as ( m - n )² , we get

$\rm : \implies \: y(m - n) = (m - n) {}^{2}$

$\rm : \implies y(m - n) = (m - n)(m - n)$

So

$: \implies \rm \: y = \dfrac{(m - n)(m - n)}{(m - n)}$

$: \implies \rm \: y = \dfrac{ \cancel{(m - n)}(m - n)}{ \cancel{(m - n)}}$

$: \implies \: \rm \: y = m - n$

Now put the value of y on iii eq

$\rm : \implies \: x = 2n + y$

$\rm : \implies \: x = 2n + (m - n)$

$\rm : \implies \: x = m + n$

So value of x = m + n and y = m - n