NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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Complex Equations
Lesson

If two complex numbers are equal then the real and imaginary parts are also equal. We call this equating like parts

Example 1

If $a+6i=3+6i$a+6i=3+6i, then $a=3$a=3

If $8-bi=8+7i$8bi=8+7i, then $b=-7$b=7

 

We can use this process to solve algebraic problems involving complex numbers.

Example 2

Find $x,y$x,y if $\left(3+2i\right)^2-3\left(x+iy\right)=x+iy$(3+2i)23(x+iy)=x+iy

  $\left(3+2i\right)^2-3\left(x+iy\right)$(3+2i)23(x+iy) $=$= $x+iy$x+iy
  $9+12i-4-3x-3yi$9+12i43x3yi $=$= $x+iy$x+iy
                                                             $5-3x+12i-3yi$53x+12i3yi $=$= $x+iy$x+iy
  $(5-3x)+i(12-3y)$(53x)+i(123y) $=$= $x+iy$x+iy
  Therefore:    
  $5-3x$53x $=$= $x$x
  $5$5 $=$= $4x$4x
  $x$x $=$= $\frac{5}{4}$54
       
  and    
  $12-3y$123y $=$= $y$y
  $12$12 $=$= $4y$4y
  $y$y $=$= $3$3

 

 

 

Example 3

Find $x,y$x,y if $\frac{x}{1-i}+\frac{y}{4+3i}=2-4i$x1i+y4+3i=24i

$\frac{x}{1-i}+\frac{y}{4+3i}$x1i+y4+3i $=$= $2-4i$24i
$\frac{x\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}+\frac{y\left(4-3i\right)}{\left(4+3i\right)\left(4-3i\right)}$x(1+i)(1i)(1+i)+y(43i)(4+3i)(43i) $=$= $2-4i$24i
$\frac{x+xi}{1+1}+\frac{4y-3yi}{16+9}$x+xi1+1+4y3yi16+9 $=$= $2-4i$24i
$\frac{x+xi}{2}+\frac{4y-3yi}{25}$x+xi2+4y3yi25 $=$= $2-4i$24i
$\frac{25x+25xi}{50}+\frac{8y-6yi}{50}$25x+25xi50+8y6yi50 $=$= $2-4i$24i
$25x+25xi+8y-6yi$25x+25xi+8y6yi $=$= $100-200i$100200i
$\left(25x+8y\right)+i\left(25x-6y\right)$(25x+8y)+i(25x6y) $=$= $100-200i$100200i
Therefore    
$25x+8y$25x+8y $=$= $100$100 (1)
$25x-6y$25x6y $=$= $-200$200 (2)

By equating like parts we can then

get simultaneous equations to solve

   
(1)-(2)                                          $14y$14y $=$= $300$300 
$y$y $=$= $\frac{300}{14}$30014
       Sub $y=\frac{300}{14}$y=30014 into (1)                         $25x+8\times\frac{300}{14}$25x+8×30014 $=$= $100$100
$25x$25x $=$= $\frac{-500}{7}$5007
$x$x $=$= $-\frac{20}{7}$207

More Worked Examples

QUESTION 1

Given that $5a+12i=10-3bi$5a+12i=103bi:

  1. find the value of $a$a

  2. find the value of $b$b

QUESTION 2

Find the value of $z$z if $\left(z+i\right)\left(3+2i\right)=20-4i$(z+i)(3+2i)=204i.

 

Outcomes

M8-9

Manipulate complex numbers and present them graphically

91577

Apply the algebra of complex numbers in solving problems

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