Solve z=1-2i/2-i

$$z = \frac { 1 - 2 i } { 2 - i }$$

Solve for z

$$z=\frac{4}{5}-\frac{3}{5}i=0.8-0.6i$$

Steps for Solving Linear Equation

Multiply both numerator and denominator of \frac{1-2i}{2-i} by the complex conjugate of the denominator, 2+i.

$$z=\frac{\left(1-2i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$z=\frac{\left(1-2i\right)\left(2+i\right)}{2^{2}-i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$z=\frac{\left(1-2i\right)\left(2+i\right)}{5}$$

Multiply complex numbers 1-2i and 2+i like you multiply binomials.

$$z=\frac{1\times 2+i-2i\times 2-2i^{2}}{5}$$

By definition, i^{2} is -1.

$$z=\frac{1\times 2+i-2i\times 2-2\left(-1\right)}{5}$$

Do the multiplications in 1\times 2+i-2i\times 2-2\left(-1\right).

$$z=\frac{2+i-4i+2}{5}$$

Combine the real and imaginary parts in 2+i-4i+2.

$$z=\frac{2+2+\left(1-4\right)i}{5}$$

Do the additions in 2+2+\left(1-4\right)i.

$$z=\frac{4-3i}{5}$$

Divide 4-3i by 5 to get \frac{4}{5}-\frac{3}{5}i.

$$z=\frac{4}{5}-\frac{3}{5}i$$

Assign z

$$z≔\frac{4}{5}-\frac{3}{5}i$$