Question

Solve for x and y:
$\begin{gathered}
  ax + by - a + b = 0, \\
  bx - ay - a - b = 0 \\
\end{gathered} $

Answer 1

Hint: This is two variables, two equations problems. Solve for $x$ first by removing $y$ using both equations. Once we end up with the value of $x$, substitute it in either of the equations to find $y$.
Complete step by step answer:
We have been given two equations and there are two variables in each equation. We will work on these equations first to solve for the value of $x$ and then $y$. Our equations are as follows:
$ax + by - a + b = 0$ ----- (1)
$bx - ay - a - b = 0$ ----- (2)
First we will remove $y$ terms first. Multiplying (1) by $a$ and (2) by $b$ on both sides, we get
${a^2}x + aby - {a^2} + ab = 0$ ---(3)
${b^2}x - aby - ab - {b^2} = 0$ ----(4)
After taking the sum of (3) and (4), we remove the terms, we get,
$\begin{gathered}
   \Rightarrow ({a^2} + {b^2})x - ({a^2} + {b^2}) = 0 \\
   \Rightarrow x = \dfrac{{({a^2} + {b^2})}}{{({a^2} + {b^2})}} = 1 \\
\end{gathered} $
Now we got the value of $x$. We will substitute this value in (1), to find the value of $y$.
$\begin{gathered}
   \Rightarrow a + by - a + b = 0 \\
   \Rightarrow by = - b \\
   \Rightarrow y = - 1 \\
\end{gathered} $
This can also be pictured in a way two expressions are the equations of two lines and the solution is the intersection of two lines if plotted on coordinate axes.
Note: This problem can also be solved by finding x in terms of y using the first equation and substituting it with the second equation to find the value of y first and then x.