# Solve this pair of simultaneous equations involving complex numbers

• chwala
It still takes a while to typeset. *grumble*grumble*deletes*half*complete*post*grumble* :wink: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 keyI prefer $$...the equations look like a piece of art...I will just need to work on the spacing. #### chwala Gold Member Homework Statement Solve the simultaneous equation for the complex number ##z## and ##w##,$$(1+i)z+(2-i)w=3+4iiz+(3+i)w=-1+5i$$Relevant Equations Complex numbers$$(1+i)z+(2-i)w=3+4iiz+(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+5iw=\frac {-1+3i}{3+i}w=\frac {(-1+3i)(3-i)}{(3+i)(3-i)}w=i$$There may be a different approach from this... Last edited: • PeroK I'm not sure there is anything easier. • chwala Well, I could debit something that is different/is not different but looks different and is probably not easier : Consider this a matrix equation ## Ax = y## with solution ##x = A^{-1}y##:$$
\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}
\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 $$for a new displayed equation creates a lot of vertical white spacing. And the more so with two carriage returns (another empty line !). But that's all beside the mathematical point, where you did just fine! ##\ ## • DaveE, Ibix and chwala BvU said: ##\LaTeX## wise this is quite a compact solution . It still takes a while to typeset. *grumble*grumble*deletes*half*complete*post*grumble* • • fresh_42 and BvU Which is the carriage return syntax? ...you could share a simple math example "with" and "without" carriage return then i can adopt that immediately... I think the point is that if you write Newton's second law says$$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. Thanks, i will check on that Ibix... 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 ! • chwala and Ibix BvU said: 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 ##.
chwala said:
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 $$...the equations look like a piece of art...I will just need to work on the spacing. ## doesn't bring out the art and desired neatness...particularly on fraction type of equations... • BvU Also Cramer's Rule, which I probably would have used. A slightly different take on the linear algebra of @BvU's post. • BvU and chwala DaveE said: Also Cramer's Rule, which I probably would have used. A slightly different take on the linear algebra of @BvU's post. Yap...Matrices in the making... chwala said: I prefer$$ ...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 .

• BvU and chwala