New Zealand
Level 8 - NCEA Level 3

# 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)2−3(x+iy) $=$= $x+iy$x+iy $9+12i-4-3x-3yi$9+12i−4−3x−3yi $=$= $x+iy$x+iy $5-3x+12i-3yi$5−3x+12i−3yi $=$= $x+iy$x+iy $(5-3x)+i(12-3y)$(5−3x)+i(12−3y) $=$= $x+iy$x+iy Therefore: $5-3x$5−3x $=$= $x$x $5$5 $=$= $4x$4x $x$x $=$= $\frac{5}{4}$54​ and $12-3y$12−3y $=$= $y$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

 $\frac{x}{1-i}+\frac{y}{4+3i}$x1−i​+y4+3i​ $=$= $2-4i$2−4i $\frac{x\left(1+i\right)}{\left(1-i\right)\left(1+i\right)}+\frac{y\left(4-3i\right)}{\left(4+3i\right)\left(4-3i\right)}$x(1+i)(1−i)(1+i)​+y(4−3i)(4+3i)(4−3i)​ $=$= $2-4i$2−4i $\frac{x+xi}{1+1}+\frac{4y-3yi}{16+9}$x+xi1+1​+4y−3yi16+9​ $=$= $2-4i$2−4i $\frac{x+xi}{2}+\frac{4y-3yi}{25}$x+xi2​+4y−3yi25​ $=$= $2-4i$2−4i $\frac{25x+25xi}{50}+\frac{8y-6yi}{50}$25x+25xi50​+8y−6yi50​ $=$= $2-4i$2−4i $25x+25xi+8y-6yi$25x+25xi+8y−6yi $=$= $100-200i$100−200i $\left(25x+8y\right)+i\left(25x-6y\right)$(25x+8y)+i(25x−6y) $=$= $100-200i$100−200i Therefore $25x+8y$25x+8y $=$= $100$100 (1) $25x-6y$25x−6y $=$= $-200$−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