Question

Solve for x and y: $mx - ny = {m^2} + {n^2}$,$x - y = 2n$

Answer 1

Hint: First, we will see there are a total of two equations given in the problem.
One equation is in degree two and the other equation is in degree one and also the straight line which is called the linear degree equation.
Try to convert the one-degree equation and apply the variables into another equation to get the values of x and y.
Take $mx - ny = {m^2} + {n^2}$as equation $(1)$ and $x - y = 2n$ as equation $(2)$.

Complete step-by-step solution:
Now take the second equation which is given in the question $x - y = 2n$.
Now try to convert the equating into one variable in the terms of the others, thus we get $x - y = 2n \Rightarrow x = 2n + y$ (value of the x is fixed and y is converted in the equation)
Thus, we have the second equation as $x = 2n + y$.
Let us substitute the second equation in the given first equation we get $mx - ny = {m^2} + {n^2} \Rightarrow m(2n + y) - ny = {m^2} + {n^2}$ (the value of the x is applied).
Try to simplify further we get $m(2n + y) - ny = {m^2} + {n^2} \Rightarrow 2mn + my - ny = {m^2} + {n^2}$(applying the terms into the equation on the left-hand side).
Taking the common term Y on the left-hand side and also replace the constant on the right-hand side
Hence, we get $2mn + my - ny = {m^2} + {n^2} \Rightarrow y(m - n) = {m^2} + {n^2} - 2mn$.
On the right side of the equation, we have to the formula of ${(a - b)^2}$ applying that we get $ \Rightarrow y(m - n) = {(m - n)^2}$
Now we are going to cancel the common terms thus $ \Rightarrow y = (m - n)$ which is the value of y.
Substitute the value of y in the second equation (converted) above we get,
$x = 2n + y \Rightarrow x = 2n + m - n$ solving further we get $x = m + n$.
Therefore, the values of x and y are $x = m + n$ and $y = (m - n)$.

Note: We are also able to solve this problem by converting equation one into one variable term as we did in the second equation.
After finding the one variable in equation one, we can substitute that into the given second equation and we can simplify the values of x and y in the same method
Also, the values of x and y are the same in both methods.