<p>If <span>\((a + b) - i(3a + 2b) = 5 + 2i\)</span>, then find <span>\(a\)</span> and <span>\(b\)</span>.</p>
Step-by-Step Solution
Key Concept: Equate real and imaginary parts separately: if z₁ = z₂, then Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂). Note the negative sign before i in the imaginary part.
<p><strong>Step 1:</strong> Write the given equation in standard form.</p><p>$(a + b) - i(3a + 2b) = 5 + 2i$</p><p>This means: $(a + b) + i[-(3a + 2b)] = 5 + 2i$</p><p><strong>Step 2:</strong> Equate real and imaginary parts separately.</p><p>Real part: $a + b = 5$ ... (1)</p><p>Imaginary part: $-(3a + 2b) = 2$</p><p>Therefore: $3a + 2b = -2$ ... (2)</p><p><strong>Step 3:</strong> Solve the system of linear equations.</p><p>From equation (1): $b = 5 - a$</p><p>Substitute into equation (2):</p><p>$3a + 2(5 - a) = -2$</p><p>$3a + 10 - 2a = -2$</p><p>$a = -12$</p><p><strong>Step 4:</strong> Find b.</p><p>$b = 5 - a = 5 - (-12) = 17$</p><p><strong>Verification:</strong> $(a + b) - i(3a + 2b) = (-12 + 17) - i(3(-12) + 2(17)) = 5 - i(-36 + 34) = 5 - i(-2) = 5 + 2i$ ✓</p><p>∴ <strong>Answer: a = -12, b = 17</strong></p>
Correct Answer: a = -12, b = 17