Solve {(3+4i)}z=|4+3i|

$${( 3 + 4 i )} z = | 4 + 3 i |$$

Solve for z

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

Steps for Solving Linear Equation

The modulus of a complex number a+bi is \sqrt{a^{2}+b^{2}}. The modulus of 4+3i is 5.

$$\left(3+4i\right)z=5$$

Divide both sides by 3+4i.

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

Multiply both numerator and denominator of \frac{5}{3+4i} by the complex conjugate of the denominator, 3-4i.

$$z=\frac{5\left(3-4i\right)}{\left(3+4i\right)\left(3-4i\right)}$$

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

$$z=\frac{5\left(3-4i\right)}{3^{2}-4^{2}i^{2}}$$

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

$$z=\frac{5\left(3-4i\right)}{25}$$

Multiply 5 times 3-4i.

$$z=\frac{5\times 3+5\times \left(-4i\right)}{25}$$

Do the multiplications in 5\times 3+5\times \left(-4i\right).

$$z=\frac{15-20i}{25}$$

Divide 15-20i by 25 to get \frac{3}{5}-\frac{4}{5}i.

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