Question:
<p>The solution of the pair of equations x + y = a + b and ax - by = a<sup>2</sup> - b<sup>2</sup> is:</p>
<p style="display:inline">x = -a, y = b</p>
<p style="display:inline">x = b, y = a</p>
<p style="display:inline">x = a, y = b</p>
<p style="display:inline">x = a, y = -b</p>
Step-by-Step Solution
Key Concept: Solve the system of linear equations by isolating one variable in the first equation and substituting its expression into the second to reduce it to a single-variable equation.
<p>The given equations are<br />
x + y = a + b ...(i)<br />
ax - by = a<sup>2</sup> - b<sup>2</sup> ...(ii)<br />
From (i)<br />
y = a + b - x<br />
Substituting y = a + b - x in (ii), we get<br />
ax - b(a + b - x) = a<sup>2</sup> - b<sup>2</sup><br />
<span class="math-tex">$\Rightarrow$</span> ax - ab - b<sup>2</sup> + bx = a<sup>2</sup> - b<sup>2</sup><br />
<span class="math-tex">$\Rightarrow$</span> x = <span class="math-tex">$\frac{a^2+a b}{a+b}$</span> = a<br />
Now, substitute x = a in (i) to get<br />
a + y = a + b<br />
<span class="math-tex">$\Rightarrow$</span> y = b<br />
Hence, x = a and y = b.</p>
Correct Answer: C