Question

If $ax + by = {a^2} - {b^2}{\text{ and }}bx + ay = 0$ , find the value of $\left( {x + y} \right)$ .

Answer 1

Hint: In order to solve the given question using the equations given, try to carry out some operation within the statement and find some coefficient of $\left( {x + y} \right)$ from them. Further use the algebraic formulas to do so.

Complete step-by-step answer:

Given equations are $ax + by = {a^2} - {b^2}{\text{ and }}bx + ay = 0$

In order to find some factor or coefficient of term $\left( {x + y} \right)$ let us add both the equations and proceed.

$ \Rightarrow \left( {ax + by} \right) + \left( {bx + ay} \right) = {a^2} - {b^2} + 0$

Simplifying further in order to find common terms
$
   \Rightarrow ax + bx + ay + by + = {a^2} - {b^2} \\
   \Rightarrow x\left( {a + b} \right) + y\left( {a + b} \right) = {a^2} - {b^2} \\
   \Rightarrow \left( {x + y} \right)\left( {a + b} \right) = {a^2} - {b^2} \\
 $

As we know the formula for algebraic theorem on RHS ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$

Using the given formula we proceed to find the value of $\left( {x + y} \right)$
$
   \Rightarrow \left( {x + y} \right)\left( {a + b} \right) = {a^2} - {b^2} \\
   \Rightarrow \left( {x + y} \right) = \dfrac{{{a^2} - {b^2}}}{{\left( {a + b} \right)}} \\
   \Rightarrow \left( {x + y} \right) = \dfrac{{\left( {a + b} \right)\left( {a - b} \right)}}{{\left( {a + b} \right)}} \\
   \Rightarrow \left( {x + y} \right) = \left( {a - b} \right) \\
 $

Hence, the value of $\left( {x + y} \right) = \left( {a - b} \right)$

Note: This question can also be solved by finding the individual values of x and y in terms of “a” and “b” separately and then the final result, but that would be a very long process. In order to solve such a question with more number of unknown variables and where some relation can be identified between the never try to find the individual values of unknown terms to find the final result. Simply try to manipulate the equations.