Solve (2+i)z=1+i

$$( 2 + i ) z = 1 + i$$

Solve for z

$$z=\frac{3}{5}+\frac{1}{5}i=0.6+0.2i$$

Steps for Solving Linear Equation

Divide both sides by 2+i.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the real and imaginary parts in 2-i+2i+1.

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

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

$$z=\frac{3+i}{5}$$

Divide 3+i by 5 to get \frac{3}{5}+\frac{1}{5}i.

$$z=\frac{3}{5}+\frac{1}{5}i$$