1.3 Given: \( (a+b i)(3-4 i)=-6-17 i \) Solve \( a+b i \) Given: \( 5 x^{2} i-4=3 y i^{4}+y^{2} i^{2}-y(x i+i) \) Solve \( x \) and \( y \) where \( y \in N \).
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To solve the equation \( (a+b i)(3-4 i)=-6-17 i \), first expand the left side: \[ (a + bi)(3 - 4i) = 3a - 4ai + 3bi + 4b = (3a + 4b) + (-4a + 3b)i. \] Now equate the real and imaginary parts: 1. \( 3a + 4b = -6 \) 2. \( -4a + 3b = -17 \) Solving this system will give the values of \( a \) and \( b \). For the second equation: \[ 5xi - 4 = 3yi(-1) + y^2(-1) - y(xi + i) \] This simplifies to: \[ 5xi - 4 = -3yi - y^2 - yxi - yi. \] Combining like terms gives: \[ 5xi + (yxi + 3yi + yi) = -4 + y^2. \] This leads us to find values of \( x \) and \( y \) by equating coefficients. Since \( y \in N \), plug in natural numbers for \( y \) to find suitable \( x \). Overall, solving the systems will provide \( a, b, x, \) and \( y \).