Solve frac{z_{1}}{z_{2}}=(2-3i)*(2+i)^2/15-5i

$$\frac { z _ { 1 } } { z _ { 2 } } = ( 2 - 3 i ) \times \frac { ( 2 + i ) ^ { 2 } } { 15 - 5 i }$$

Solve for z_1

$$z_{1}=\left(\frac{11}{10}+\frac{3}{10}i\right)z_{2}$$

$$z_{2}\neq 0$$

Solution Steps

Multiply both sides of the equation by z_{2}.

$$z_{1}=\left(2-3i\right)\left(\frac{3}{50}+\frac{1}{50}i\right)z_{2}\left(2+i\right)^{2}$$

Multiply 2-3i and \frac{3}{50}+\frac{1}{50}i to get \frac{9}{50}-\frac{7}{50}i.

$$z_{1}=\left(\frac{9}{50}-\frac{7}{50}i\right)z_{2}\left(2+i\right)^{2}$$

Calculate 2+i to the power of 2 and get 3+4i.

$$z_{1}=\left(\frac{9}{50}-\frac{7}{50}i\right)z_{2}\left(3+4i\right)$$

Multiply \frac{9}{50}-\frac{7}{50}i and 3+4i to get \frac{11}{10}+\frac{3}{10}i.

$$z_{1}=\left(\frac{11}{10}+\frac{3}{10}i\right)z_{2}$$

Solve for z_2

$$z_{2}=\left(\frac{11}{13}-\frac{3}{13}i\right)z_{1}$$

$$z_{1}\neq 0$$

Steps for Solving Linear Equation

Variable z_{2} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by z_{2}.

$$z_{1}=\left(2-3i\right)\left(\frac{3}{50}+\frac{1}{50}i\right)z_{2}\left(2+i\right)^{2}$$

Multiply 2-3i and \frac{3}{50}+\frac{1}{50}i to get \frac{9}{50}-\frac{7}{50}i.

$$z_{1}=\left(\frac{9}{50}-\frac{7}{50}i\right)z_{2}\left(2+i\right)^{2}$$

Calculate 2+i to the power of 2 and get 3+4i.

$$z_{1}=\left(\frac{9}{50}-\frac{7}{50}i\right)z_{2}\left(3+4i\right)$$

Multiply \frac{9}{50}-\frac{7}{50}i and 3+4i to get \frac{11}{10}+\frac{3}{10}i.

$$z_{1}=\left(\frac{11}{10}+\frac{3}{10}i\right)z_{2}$$

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

$$\left(\frac{11}{10}+\frac{3}{10}i\right)z_{2}=z_{1}$$

Divide both sides by \frac{11}{10}+\frac{3}{10}i.

$$\frac{\left(\frac{11}{10}+\frac{3}{10}i\right)z_{2}}{\frac{11}{10}+\frac{3}{10}i}=\frac{z_{1}}{\frac{11}{10}+\frac{3}{10}i}$$

Dividing by \frac{11}{10}+\frac{3}{10}i undoes the multiplication by \frac{11}{10}+\frac{3}{10}i.

$$z_{2}=\frac{z_{1}}{\frac{11}{10}+\frac{3}{10}i}$$

Divide z_{1} by \frac{11}{10}+\frac{3}{10}i.

$$z_{2}=\left(\frac{11}{13}-\frac{3}{13}i\right)z_{1}$$

Variable z_{2} cannot be equal to 0.

$$z_{2}=\left(\frac{11}{13}-\frac{3}{13}i\right)z_{1}\text{, }z_{2}\neq 0$$