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Given that [tex]\( x = 3 + 8i \)[/tex] and [tex]\( y = 7 - i \)[/tex], match the equivalent expressions.

A. [tex]\( -15 + 19i \)[/tex]
B. [tex]\( -29 - 53i \)[/tex]
C. [tex]\( -8 \)[/tex]
D. [tex]\( 41i \)[/tex]
E. [tex]\( 58 + 106i \)[/tex]

1. [tex]\( -x \cdot y \longrightarrow \square \)[/tex]

2. [tex]\( x \cdot 2y \longrightarrow \square \)[/tex]

3. [tex]\( 2x - 3y \longrightarrow \square \)[/tex]



Answer :

GinnyAnswer
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]

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