Question

Solve the following pair of linear equations:
$px + qy = p - q \\\ qx - py = p + q \\\$

Answer 1

Hint: In this question we use the addition method of linear equations which means two equations are added together to eliminate one of the variables. Use this method to solve the equations.

Complete step-by-step answer:
According to the question , two linear equations are given i.e.
$px + qy = p - q \\\ qx - py = p + q \\\$
Where,
$px + qy = p - q.....(1) \\\ qx - py = p + q.......(2) \\\$
To make calculation easy if we multiply $eq(1)$ by $p$ and $eq(2)$ by $q$, we get
${p^2}x + pqy = {p^2} - pq......(3) \\\ {q^2}x - pqy = pq + {q^2}.......(4) \\\$
Now, add $eq(3)$ and $eq(4)$ we get,
${p^2}x + {q^2}x = {p^2} + {q^2} \\\ \\\$
Now ,taking $x$ as common on $L.H.S.$ we get
$x({p^2} + {q^2}) = {p^2} + {q^2} \\\ x = 1 \\\$
Putting the value of $x$ in $eq(1)$we get,
$\Rightarrow p.1 + qy = p - q \\\ \Rightarrow qy = p - q - p \\\ \Rightarrow qy = - q \\\ \Rightarrow y = - 1 \\\ \therefore x = 1,y = - 1 \\\$

Note: In such types of questions when we are adding two linear equations , we should keep in mind the elimination method like multiplying one of the equations with such a variable that while adding one variable will be eliminated.