Question

Solve for x and y:
$\begin{align}
  & mx-ny={{m}^{2}}+{{n}^{2}}; \\
 & x+y=2m \\
\end{align}$

Answer 1

Hint: To solve the given pair of simultaneous equations, multiply the second equation by “n” and then add this new equation with the first equation. After addition, you will find that y is eliminated and then solve for x. Now substitute this value of x in the second equation in the given pair of linear equations you will find the value of y.

Complete step-by-step answer:
The given pair of simultaneous equations is:
$\begin{align}
  & mx-ny={{m}^{2}}+{{n}^{2}}; \\
 & x+y=2m \\
\end{align}$
Multiplying the second equation of the above pair of equations by “n” we get,
$nx+ny=2mn$ ………. Eq. (1)
$mx-ny={{m}^{2}}+{{n}^{2}}$ ……… Eq. (2)
Adding eq. (1) and eq. (2) we get,
$\begin{matrix}
  nx+ny=2mn \\
  \dfrac{+\left( mx-ny={{m}^{2}}+{{n}^{2}} \right)}{\left( n+m \right)x=2mn+{{m}^{2}}+{{n}^{2}}} \\
\end{matrix}$
Solving the above resulting equation we get,
$\begin{align}
  & \left( n+m \right)x=2mn+{{m}^{2}}+{{n}^{2}} \\
 & \Rightarrow \left( n+m \right)x={{\left( n+m \right)}^{2}} \\
\end{align}$
In the above equation $\left( n+m \right)$ will be cancelled out on both the sides of the equation.
$x=n+m$
Substituting the above value of x in one of the pair of given equations we get,
$\begin{align}
  & x+y=2m \\
 & \Rightarrow n+m+y=2m \\
 & \Rightarrow y=m-n \\
\end{align}$
From the above equations, the value of x and y is:
$\begin{align}
  & x=n+m; \\
 & y=m-n \\
\end{align}$

Note: You can verify the solutions that we have got above by satisfying the equations given in the questions.
The value of x and y that we have obtained above is:
$\begin{align}
  & x=n+m; \\
 & y=m-n \\
\end{align}$
 Substituting the above values of x and y in any one of the given pair of equations:
$\begin{align}
  & x+y=2m \\
 & \Rightarrow n+m+m-n=2m \\
\end{align}$
In the above equation, “n” in the left hand side of the equation will be cancelled out and the expression that we are left with look like:
$2m=2m$
As you can see from the above equation that L.H.S = R.H.S so we have shown that the values of x and y that we have got have satisfied one of the pairs of given equations.
Similarly, you can check that the values of x and y are satisfying the other pair of given equations also.