Let's solve the expressions one by one, given that [tex]\( x = 3 + 8i \)[/tex] and [tex]\( y = 7 - i \)[/tex].
1. Evaluating the expression [tex]\( -x \cdot y \)[/tex]:
[tex]\[
-x \cdot y = -(3 + 8i) \cdot (7 - i)
\][/tex]
After performing the complex multiplication and subsequent negation, we get:
[tex]\[
-x \cdot y = -29 - 53i.
\][/tex]
2. Evaluating the expression [tex]\( x \cdot 2y \)[/tex]:
[tex]\[
x \cdot 2y = (3 + 8i) \cdot 2(7 - i) = (3 + 8i) \cdot (14 - 2i)
\][/tex]
After performing the complex multiplication, we get:
[tex]\[
x \cdot 2y = 58 + 106i.
\][/tex]
3. Evaluating the expression [tex]\( 2x - 3y \)[/tex]:
[tex]\[
2x = 2 \cdot (3 + 8i) = 6 + 16i
\][/tex]
[tex]\[
3y = 3 \cdot (7 - i) = 21 - 3i
\][/tex]
[tex]\[
2x - 3y = (6 + 16i) - (21 - 3i) = 6 + 16i - 21 + 3i = -15 + 19i.
\][/tex]
So the matching of the equivalent expressions is as follows:
[tex]\[
-x \cdot y \longrightarrow -29 - 53i
\][/tex]
[tex]\[
x \cdot 2 y \longrightarrow 58 + 106i
\][/tex]
[tex]\[
2 x-3 y \longrightarrow -15 + 19i
\][/tex]
To summarize:
[tex]\[
-x \cdot y \longrightarrow -29 - 53i
\][/tex]
[tex]\[
x \cdot 2 y \longrightarrow 58 + 106i
\][/tex]
[tex]\[
2 x-3 y \longrightarrow -15 + 19i
\][/tex]
