To solve these expressions given [tex]\( x = 3 + 8i \)[/tex] and [tex]\( y = 7 - i \)[/tex], let's break down each calculation step-by-step:
1. Expression: [tex]\( x \cdot 2 \cdot y \)[/tex]
To compute this, let's perform the multiplication as follows:
- First, compute [tex]\( 2 \cdot y \)[/tex]:
[tex]\[
2 \cdot (7 - i) = 14 - 2i
\][/tex]
- Now, multiply [tex]\( x \)[/tex] by this result:
[tex]\[
(3 + 8i) \cdot (14 - 2i)
\][/tex]
Using the distributive property (FOIL method):
[tex]\[
= 3 \cdot 14 + 3 \cdot (-2i) + 8i \cdot 14 - 8i \cdot 2i
\][/tex]
[tex]\[
= 42 - 6i + 112i - 16i^2
\][/tex]
Recall [tex]\( i^2 = -1 \)[/tex], so:
[tex]\[
= 42 - 6i + 112i - 16(-1)
\][/tex]
[tex]\[
= 42 - 6i + 112i + 16
\][/tex]
Combine the real and imaginary parts:
[tex]\[
= (42 + 16) + (-6i + 112i)
\][/tex]
[tex]\[
= 58 + 106i
\][/tex]
2. Expression: [tex]\( -5x + y \)[/tex]
To compute this, let's perform the addition and multiplication as follows:
- First, compute [tex]\( -5x \)[/tex]:
[tex]\[
-5 \cdot (3 + 8i) = -15 - 40i
\][/tex]
- Now, add [tex]\( y \)[/tex] to this result:
[tex]\[
-15 - 40i + (7 - i)
\][/tex]
Combine the real and imaginary parts:
[tex]\[
= (-15 + 7) + (-40i - i)
\][/tex]
[tex]\[
= -8 - 41i
\][/tex]
Now, we can pair the expressions with their corresponding results:
- For [tex]\( x \cdot 2 \cdot y \)[/tex], the result is [tex]\( 58 + 106i \)[/tex].
- For [tex]\( -5x + y \)[/tex], the result is [tex]\( -8 - 41i \)[/tex].
So, the pairing will be:
- [tex]\( x \cdot 2 \cdot y \longrightarrow 58 + 106i \)[/tex]
- [tex]\( -5 x + y \longrightarrow -8 - 41i \)[/tex]
