<p>Find the real numbers \(x\) and \(y\) if \((x - iy)(3 + 5i)\) is the conjugate of \(-6 - 24i\).</p>
Step-by-Step Solution
Key Concept: The conjugate of -6 - 24i is -6 + 24i. Expand (x - iy)(3 + 5i) and equate real and imaginary parts separately to this conjugate.
<p><strong>Step 1:</strong> Find the conjugate of -6 - 24i, which is <strong>-6 + 24i</strong>.</p><p><strong>Step 2:</strong> Expand (x - iy)(3 + 5i):<br/>(x - iy)(3 + 5i) = 3x + 5ix - 3iy - 5i²y<br/>= 3x + 5ix - 3iy + 5y (since i² = -1)<br/>= <strong>(3x + 5y) + i(5x - 3y)</strong></p><p><strong>Step 3:</strong> Equate to the conjugate -6 + 24i by comparing real and imaginary parts:<br/>Real part: 3x + 5y = -6 ... (1)<br/>Imaginary part: 5x - 3y = 24 ... (2)</p><p><strong>Step 4:</strong> Solve the system. Multiply (1) by 3 and (2) by 5:<br/>9x + 15y = -18<br/>25x - 15y = 120</p><p><strong>Step 5:</strong> Add the equations: 34x = 102, so <strong>x = 3</strong></p><p><strong>Step 6:</strong> Substitute x = 3 into equation (1):<br/>3(3) + 5y = -6<br/>9 + 5y = -6<br/>5y = -15<br/><strong>y = -3</strong></p><p>∴ Answer: <strong>x = 3, y = -3</strong></p>
Correct Answer: x = 3, y = -3