<p>Find the number of integral solutions of \(\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{7}\).</p>
Step-by-Step Solution
Key Concept: Rearrange the equation to isolate one variable as a function of the other, then determine when the expression yields an integer. Specifically, manipulate algebraically to get y = 7x/(x-7), then find which integer values of x make y an integer.
<p><strong>Step 1:</strong> Start with $\frac{1}{x} + \frac{1}{y} = \frac{1}{7}$</p><p><strong>Step 2:</strong> Rearrange: $\frac{1}{y} = \frac{1}{7} - \frac{1}{x} = \frac{x-7}{7x}$</p><p><strong>Step 3:</strong> Therefore: $y = \frac{7x}{x-7}$</p><p><strong>Step 4:</strong> For y to be an integer, $(x-7)$ must divide $7x$. Rewrite: $y = \frac{7(x-7) + 49}{x-7} = 7 + \frac{49}{x-7}$</p><p><strong>Step 5:</strong> For y to be an integer, $(x-7)$ must divide 49. The divisors of 49 are: $\pm 1, \pm 7, \pm 49$</p><p><strong>Step 6:</strong> This gives $x - 7 \in \{-49, -7, -1, 1, 7, 49\}$</p><p><strong>Step 7:</strong> So $x \in \{-42, 0, 6, 8, 14, 56\}$. Since $x \neq 0$, we have $x \in \{-42, 6, 8, 14, 56\}$</p><p><strong>Step 8:</strong> Computing corresponding y-values:</p><ul><li>$x = -42$: $y = 7 + \frac{49}{-49} = 6$ ✓</li><li>$x = 6$: $y = 7 + \frac{49}{-1} = -42$ ✓</li><li>$x = 8$: $y = 7 + \frac{49}{1} = 56$ ✓</li><li>$x = 14$: $y = 7 + \frac{49}{7} = 14$ ✓</li><li>$x = 56$: $y = 7 + \frac{49}{49} = 8$ ✓</li></ul><p>∴ Answer: <strong>5 integral solutions</strong></p>
Correct Answer: 5