Let's go through the process of solving for [tex]\(2x - 3y\)[/tex] step-by-step, given [tex]\(x = 3 + 8i\)[/tex] and [tex]\(y = 7 - i\)[/tex].
1. Multiply [tex]\(x\)[/tex] by 2:
[tex]\[
2x = 2 \cdot (3 + 8i) = 6 + 16i
\][/tex]
2. Multiply [tex]\(y\)[/tex] by 3:
[tex]\[
3y = 3 \cdot (7 - i) = 21 - 3i
\][/tex]
3. Subtract [tex]\(3y\)[/tex] from [tex]\(2x\)[/tex]:
[tex]\[
2x - 3y = (6 + 16i) - (21 - 3i)
\][/tex]
4. Distribute the subtraction:
[tex]\[
2x - 3y = (6 - 21) + (16i + 3i)
\][/tex]
5. Combine the real and imaginary parts:
[tex]\[
2x - 3y = -15 + 19i
\][/tex]
Therefore, the equivalent expression for [tex]\(2x - 3y\)[/tex] is [tex]\(-15 + 19i\)[/tex]. So,
[tex]\[
2x - 3y \longrightarrow -15 + 19i
\][/tex]
