11TH•Mathematics•Complex Numbers•Question 5

Solution

If $z_1=x+yi$ and $z_2=a+bi$, find $x,y,a,b$ such that $z_1+z_2=10+4i$ and $z_1-z_2=6+2i$.

Given:

z_1+z_2=10+4i,\qquad z_1-z_2=6+2i

Add the equations:

\begin{aligned} (z_1+z_2)+(z_1-z_2)&=(10+4i)+(6+2i) \\ 2z_1&=16+6i \\ z_1&=8+3i \end{aligned}

Now substitute into $z_1+z_2=10+4i$ :

\begin{aligned} (8+3i)+z_2&=10+4i \\ z_2&=2+i \end{aligned}

So:

z_1=x+yi=8+3i\Rightarrow x=8,\ y=3

z_2=a+bi=2+i\Rightarrow a=2,\ b=1

\boxed{x=8,\ y=3,\ a=2,\ b=1}

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