Question

Solve for x and y using substitution method:
$x+y=a-b$ , $ax-by={{a}^{2}}+{{b}^{2}}$

Answer 1

Hint: According to the question we need to solve the two linear equations using a substitution method. Here in the given equations the unknown variables are x and y. Now, we need to find the value of one variable using both the equations and then find the value of another variable.

Complete step by step answer:
According to the questions two equations are given and we are asked to solve the two linear equations using the specified substitution method.
Now, in this what we need to do is we need to find the value of one variable from one equation and then substitute the value of that variable in another equation and then using the value gained we will get the value of another variable also.
Now, from (a) and finding the value of x using first equation we get, $x=a-b-y$ and putting this in second equation we get $a\left( a-b-y \right)-by={{a}^{2}}+{{b}^{2}}$ , now simplifying this we get,
$\begin{align}
  & a\left( a-b-y \right)-by={{a}^{2}}+{{b}^{2}} \\
 & \Rightarrow {{a}^{2}}-ab-ay-by={{a}^{2}}+{{b}^{2}} \\
 & \Rightarrow -ab-{{b}^{2}}=\left( a+b \right)y \\
 & \Rightarrow y=-\dfrac{\left( ab+{{b}^{2}} \right)}{a+b} \\
 & \Rightarrow y=-b \\
\end{align}$
Now, putting this value of y in $x=a-b-y$ we get,
$\begin{align}
  & x=a-b+b \\
 & \Rightarrow x=a \\
\end{align}$.
Therefore, solving the above equations, we get the following values of x and y:
$x=a$ and $y=-b$ .

Note: In such questions, initially we need to solve the equations by making the same coefficients so we need to take care of the mistakes and do the calculations properly and also, we need to remember the process name as we can solve the given linear equations by using various methods like eliminations method etc.