Complex Algebra/Examples/2x - 3iy + 4ix - 2y - 5 - 10i = (x + y + 2) - (y - x + 3)i
Jump to navigation
Jump to search
Example of Complex Algebra
Let:
- $2 x - 3 i y + 4 i x - 2 y - 5 - 10 i = \paren {x + y + 2} - \paren {y - x + 3} i$
Then:
| \(\ds x\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds -2\) |
Proof
Separating the real parts from the imaginary parts:
| \(\ds 2 x - 2 y - 5\) | \(=\) | \(\ds x + y + 2\) | ||||||||||||
| \(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 3 y + 7\) |
| \(\ds -3 i y + 4 i x - 10 i\) | \(=\) | \(\ds \paren {y - x + 3} i\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -3 y + 4 x - 10\) | \(=\) | \(\ds y - x + 3\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 5 x - 4 y\) | \(=\) | \(\ds 13\) |
Substituting for $x$ from $(1)$ into $(2)$:
| \(\ds 5 \paren {3 y + 7} - 4 y\) | \(=\) | \(\ds 13\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 15 y + 35 - 4 y\) | \(=\) | \(\ds 13\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 11 y\) | \(=\) | \(\ds -22\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -2\) | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 3 \times {-2} + 7\) | substituting for $y$ from $(3)$ into $(1)$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $57$