Question

let x = 2 + 8i and xy = -2 + 26i. what is y? y =

Explanation:

Step1: Isolate $y$

$y = \frac{-2 + 26i}{2 + 8i}$

Step2: Multiply by conjugate

Multiply numerator/denominator by $2-8i$:
$y = \frac{(-2 + 26i)(2 - 8i)}{(2 + 8i)(2 - 8i)}$

Step3: Expand numerator

Calculate product of numerator terms:
$(-2)(2) + (-2)(-8i) + 26i(2) + 26i(-8i) = -4 + 16i + 52i - 208i^2$
Since $i^2=-1$, this becomes:
$-4 + 68i - 208(-1) = -4 + 68i + 208 = 204 + 68i$

Step4: Expand denominator

Calculate product of denominator terms:
$2^2 - (8i)^2 = 4 - 64i^2 = 4 - 64(-1) = 4 + 64 = 68$

Step5: Simplify fraction

Divide numerator by denominator:
$y = \frac{204 + 68i}{68} = \frac{204}{68} + \frac{68i}{68}$

Answer:

$3 + i$