Solve 3+4i/z-2=3-i

$$\frac { 3 + 4 i } { z - 2 } = 3 - i$$

Solve for z

$$z=\frac{5}{2}+\frac{3}{2}i=2.5+1.5i$$

Steps for Solving Linear Equation

Variable z cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by z-2.

$$3+4i=\left(3-i\right)\left(z-2\right)$$

Use the distributive property to multiply 3-i by z-2.

$$3+4i=\left(3-i\right)z+\left(-6+2i\right)$$

Swap sides so that all variable terms are on the left hand side.

$$\left(3-i\right)z+\left(-6+2i\right)=3+4i$$

Subtract -6+2i from both sides.

$$\left(3-i\right)z=3+4i-\left(-6+2i\right)$$

Subtract -6+2i from 3+4i to get 9+2i.

$$\left(3-i\right)z=9+2i$$

Divide both sides by 3-i.

$$z=\frac{9+2i}{3-i}$$

Multiply both numerator and denominator of \frac{9+2i}{3-i} by the complex conjugate of the denominator, 3+i.

$$z=\frac{\left(9+2i\right)\left(3+i\right)}{\left(3-i\right)\left(3+i\right)}$$

Do the multiplications in \frac{\left(9+2i\right)\left(3+i\right)}{\left(3-i\right)\left(3+i\right)}.

$$z=\frac{25+15i}{10}$$

Divide 25+15i by 10 to get \frac{5}{2}+\frac{3}{2}i.

$$z=\frac{5}{2}+\frac{3}{2}i$$