Solve this pair of simultaneous equations involving complex numbers

[chwala]

Homework Statement

Solve the simultaneous equation for the complex number $z$ and $w$,

$$(1+i)z+(2-i)w=3+4i$$

$$iz+(3+i)w=-1+5i$$

Relevant Equations

Complex numbers

$$(1+i)z+(2-i)w=3+4i$$
$$iz+(3+i)w=-1+5i$$

ok, multiplying the first equation by$(1-i)$ and the second equation by $i$, we get,

$$2z+(1-3i)w=7+i$$
$$-z+(-1+3i)w=-5-i$$

adding the two equations, we get $z=2$,
We know that, $$iz+(3+i)w=-1+5i$$
$$⇒2i+(3+i)w=-1+5i$$
$$w=\frac {-1+3i}{3+i}$$
$$w=\frac {(-1+3i)(3-i)}{(3+i)(3-i)}$$
$$w=i$$

There may be a different approach from this...

 

[PeroK]

I'm not sure there is anything easier.

 

[BvU]

Well, I could debit something that is different/is not different but looks different and is probably not easier :
Consider this a matrix equation $Ax = y$ with solution $x = A^{-1}y$:$$
\begin{pmatrix} 1+i&2-i\\\phantom{1+}i&3+i\end {pmatrix}
\begin{pmatrix} z\\w\end {pmatrix}=\begin{pmatrix} \phantom{-}3+4i\\-1+5i\end {pmatrix}
\quad \Rightarrow \quad
\begin{pmatrix} z\\w\end {pmatrix} =
{1\over |A| }\begin{pmatrix} 3+i&-2+i\\\phantom{1}-i&\phantom{-}1+i\end {pmatrix}
\begin{pmatrix} \phantom{-}3+4i\\-1+5i\end {pmatrix}
$$With $|A| =\det A = (1+i)(3+i)-i(2-i) = 1 + 2i\$we get $$
\begin{pmatrix} z\\w\end {pmatrix} = {1\over 1 + 2i }
\begin{pmatrix} (3+i)(3+4i)+(-2+i)(-1+5i) \\ \phantom 3\ - i\phantom )(3+4i)+( \phantom{-}1+i)(-1+5i) \end {pmatrix} = {1\over 1 + 2i }
\begin{pmatrix}\phantom{-} 2+4i\\-2+{\phantom 4}i \end {pmatrix} =
\begin{pmatrix}2\\i\end {pmatrix}$$
$\LaTeX$ wise this is quite a compact solution .
Yours could also be slightly more compact if you realize that $$ followed by a carriage return and then $$ for a new displayed equation creates a lot of vertical white spacing. And the more so with two carriage returns (another empty line !).

But that's all beside the mathematical point, where you did just fine!

$\$

 

[Ibix]

$\LaTeX$ wise this is quite a compact solution .

It still takes a while to typeset. *grumble*grumble*deletes*half*complete*post*grumble*

 

[chwala]

Which is the carriage return syntax? ...you could share a simple math example "with" and "without" carriage return then i can adopt that immediately...

 

[Ibix]

I think the point is that if you write Newton's second law says $$F=ma$$ and is the definition of "force". you get:

Newton's second law says $$F=ma$$and is the definition of "force".​

But I think you're inserting a blank line (or at least a new line) before and after the LaTeX delimiters, so you get

Newton's second law says​

​

$$F=ma$$​

​

and is the definition of "force".​

...which takes up a lot more room vertically.

 

[chwala]

Thanks, i will check on that Ibix...

 

[BvU]

This time I had to grmbl grmbl

And I also lost the part I did want to keep , namely:
I had fun discovering that googling (3+i)(3+4i)+(-2+i)(-1+5i)
(copied straight from the $\TeX$ source!) gives $2+4i$
But horizontally aligning (-i)(3+4i)+(1+i)(-1+5i) was nightmarish indeed

@chwala: experiment !

 

[Mark44]

Yours could also be slightly more compact if you realize that $$ followed by a carriage return and then $$ for a new displayed equation creates a lot of vertical white spacing. And the more so with two carriage returns (another empty line !).

I rarely use $$ for this exact reason. I mostly use ##.

Which is the carriage return syntax? ...you could share a simple math example "with" and "without" carriage return then i can adopt that immediately...

Carriage return -- enter key

 

[chwala]

I prefer $$ ...the equations look like a piece of art...I will just need to work on the spacing.

## doesn't bring out the art and desired neatness...particularly on fraction type of equations...

 

[DaveE]

Also Cramer's Rule, which I probably would have used. A slightly different take on the linear algebra of @BvU's post.

 

[chwala]

Also Cramer's Rule, which I probably would have used. A slightly different take on the linear algebra of @BvU's post.

Yap...Matrices in the making...

 

[SammyS]

I prefer $$ ...the equations look like a piece of art...I will just need to work on the spacing.

## doesn't bring out the art and desired neatness...particularly on fraction type of equations...

For fractions with ## , use \dfrac for a fraction rather than \frac .