Solve 1/2t+1+1/27-i=4/z+2

$$\frac { 1 } { 2 t + 1 } + \frac { 1 } { 27 - i } = \frac { 4 } { z + 2 }$$

Solve for t

$$t=-\frac{\left(757+i\right)z+\left(-1406+2i\right)}{2\left(\left(27+i\right)z+\left(-2866+2i\right)\right)}$$

$$z\neq -2\text{ and }z\neq 106-4i$$

Steps for Solving Linear Equation

Variable t cannot be equal to -\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(z+2\right)\left(2t+1\right), the least common multiple of 2t+1,z+2.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{1}{27-i}=\left(2t+1\right)\times 4$$

Multiply both numerator and denominator of \frac{1}{27-i} by the complex conjugate of the denominator, 27+i.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{1\left(27+i\right)}{\left(27-i\right)\left(27+i\right)}=\left(2t+1\right)\times 4$$

Do the multiplications in \frac{1\left(27+i\right)}{\left(27-i\right)\left(27+i\right)}.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{27+i}{730}=\left(2t+1\right)\times 4$$

Divide 27+i by 730 to get \frac{27}{730}+\frac{1}{730}i.

$$z+2+\left(z+2\right)\left(2t+1\right)\left(\frac{27}{730}+\frac{1}{730}i\right)=\left(2t+1\right)\times 4$$

Use the distributive property to multiply z+2 by 2t+1.

$$z+2+\left(2zt+z+4t+2\right)\left(\frac{27}{730}+\frac{1}{730}i\right)=\left(2t+1\right)\times 4$$

Use the distributive property to multiply 2zt+z+4t+2 by \frac{27}{730}+\frac{1}{730}i.

$$z+2+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{27}{730}+\frac{1}{730}i\right)z+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\left(\frac{27}{365}+\frac{1}{365}i\right)=\left(2t+1\right)\times 4$$

Do the additions in 2+\left(\frac{27}{365}+\frac{1}{365}i\right).

$$z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{27}{730}+\frac{1}{730}i\right)z+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=\left(2t+1\right)\times 4$$

Combine z and \left(\frac{27}{730}+\frac{1}{730}i\right)z to get \left(\frac{757}{730}+\frac{1}{730}i\right)z.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=\left(2t+1\right)\times 4$$

Use the distributive property to multiply 2t+1 by 4.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=8t+4$$

Subtract 8t from both sides.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i-8t=4$$

Combine \left(\frac{54}{365}+\frac{2}{365}i\right)t and -8t to get \left(-\frac{2866}{365}+\frac{2}{365}i\right)t.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=4$$

Subtract \left(\frac{757}{730}+\frac{1}{730}i\right)z from both sides.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=4-\left(\frac{757}{730}+\frac{1}{730}i\right)z$$

Subtract \frac{757}{365} from both sides.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t+\frac{1}{365}i=4-\left(\frac{757}{730}+\frac{1}{730}i\right)z-\frac{757}{365}$$

Subtract \frac{1}{365}i from both sides.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t=4-\left(\frac{757}{730}+\frac{1}{730}i\right)z-\frac{757}{365}-\frac{1}{365}i$$

Do the additions in 4-\frac{757}{365}-\frac{1}{365}i.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t=-\left(\frac{757}{730}+\frac{1}{730}i\right)z+\frac{703}{365}-\frac{1}{365}i$$

Multiply -1 and \frac{757}{730}+\frac{1}{730}i to get -\frac{757}{730}-\frac{1}{730}i.

$$\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(-\frac{2866}{365}+\frac{2}{365}i\right)t=\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\frac{703}{365}-\frac{1}{365}i$$

Combine all terms containing t.

$$\left(\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)\right)t=\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\frac{703}{365}-\frac{1}{365}i$$

The equation is in standard form.

$$\left(\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)\right)t=\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\left(\frac{703}{365}-\frac{1}{365}i\right)$$

Divide both sides by \left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right).

$$\frac{\left(\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)\right)t}{\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)}=\frac{\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\left(\frac{703}{365}-\frac{1}{365}i\right)}{\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)}$$

Dividing by \left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right) undoes the multiplication by \left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right).

$$t=\frac{\left(-\frac{757}{730}-\frac{1}{730}i\right)z+\left(\frac{703}{365}-\frac{1}{365}i\right)}{\left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right)}$$

Divide \left(-\frac{757}{730}-\frac{1}{730}i\right)z+\left(\frac{703}{365}-\frac{1}{365}i\right) by \left(\frac{27}{365}+\frac{1}{365}i\right)z+\left(-\frac{2866}{365}+\frac{2}{365}i\right).

$$t=\frac{\left(-757-i\right)z+\left(1406-2i\right)}{2\left(\left(27+i\right)z+\left(-2866+2i\right)\right)}$$

Variable t cannot be equal to -\frac{1}{2}.

$$t=\frac{\left(-757-i\right)z+\left(1406-2i\right)}{2\left(\left(27+i\right)z+\left(-2866+2i\right)\right)}\text{, }t\neq -\frac{1}{2}$$

Solve for z

$$z=-\frac{2\left(\left(2866-2i\right)t+\left(703-i\right)\right)}{\left(-54-2i\right)t+\left(-757-i\right)}$$

$$t\neq -\frac{1}{2}\text{ and }t\neq -14+\frac{1}{2}i$$

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 \left(z+2\right)\left(2t+1\right), the least common multiple of 2t+1,z+2.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{1}{27-i}=\left(2t+1\right)\times 4$$

Multiply both numerator and denominator of \frac{1}{27-i} by the complex conjugate of the denominator, 27+i.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{1\left(27+i\right)}{\left(27-i\right)\left(27+i\right)}=\left(2t+1\right)\times 4$$

Do the multiplications in \frac{1\left(27+i\right)}{\left(27-i\right)\left(27+i\right)}.

$$z+2+\left(z+2\right)\left(2t+1\right)\times \frac{27+i}{730}=\left(2t+1\right)\times 4$$

Divide 27+i by 730 to get \frac{27}{730}+\frac{1}{730}i.

$$z+2+\left(z+2\right)\left(2t+1\right)\left(\frac{27}{730}+\frac{1}{730}i\right)=\left(2t+1\right)\times 4$$

Use the distributive property to multiply z+2 by 2t+1.

$$z+2+\left(2zt+z+4t+2\right)\left(\frac{27}{730}+\frac{1}{730}i\right)=\left(2t+1\right)\times 4$$

Use the distributive property to multiply 2zt+z+4t+2 by \frac{27}{730}+\frac{1}{730}i.

$$z+2+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{27}{730}+\frac{1}{730}i\right)z+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\left(\frac{27}{365}+\frac{1}{365}i\right)=\left(2t+1\right)\times 4$$

Do the additions in 2+\left(\frac{27}{365}+\frac{1}{365}i\right).

$$z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{27}{730}+\frac{1}{730}i\right)z+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=\left(2t+1\right)\times 4$$

Combine z and \left(\frac{27}{730}+\frac{1}{730}i\right)z to get \left(\frac{757}{730}+\frac{1}{730}i\right)z.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=\left(2t+1\right)\times 4$$

Use the distributive property to multiply 2t+1 by 4.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\left(\frac{54}{365}+\frac{2}{365}i\right)t+\frac{757}{365}+\frac{1}{365}i=8t+4$$

Subtract \left(\frac{54}{365}+\frac{2}{365}i\right)t from both sides.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\frac{757}{365}+\frac{1}{365}i=8t+4-\left(\frac{54}{365}+\frac{2}{365}i\right)t$$

Combine 8t and \left(-\frac{54}{365}-\frac{2}{365}i\right)t to get \left(\frac{2866}{365}-\frac{2}{365}i\right)t.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\frac{757}{365}+\frac{1}{365}i=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+4$$

Subtract \frac{757}{365} from both sides.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\frac{1}{365}i=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+4-\frac{757}{365}$$

Subtract \frac{757}{365} from 4 to get \frac{703}{365}.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt+\frac{1}{365}i=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\frac{703}{365}$$

Subtract \frac{1}{365}i from both sides.

$$\left(\frac{757}{730}+\frac{1}{730}i\right)z+\left(\frac{27}{365}+\frac{1}{365}i\right)zt=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\frac{703}{365}-\frac{1}{365}i$$

Combine all terms containing z.

$$\left(\frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t\right)z=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\frac{703}{365}-\frac{1}{365}i$$

The equation is in standard form.

$$\left(\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)\right)z=\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\left(\frac{703}{365}-\frac{1}{365}i\right)$$

Divide both sides by \frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t.

$$\frac{\left(\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)\right)z}{\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)}=\frac{\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\left(\frac{703}{365}-\frac{1}{365}i\right)}{\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)}$$

Dividing by \frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t undoes the multiplication by \frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t.

$$z=\frac{\left(\frac{2866}{365}-\frac{2}{365}i\right)t+\left(\frac{703}{365}-\frac{1}{365}i\right)}{\left(\frac{27}{365}+\frac{1}{365}i\right)t+\left(\frac{757}{730}+\frac{1}{730}i\right)}$$

Divide \left(\frac{2866}{365}-\frac{2}{365}i\right)t+\left(\frac{703}{365}-\frac{1}{365}i\right) by \frac{757}{730}+\frac{1}{730}i+\left(\frac{27}{365}+\frac{1}{365}i\right)t.

$$z=\frac{2\left(\left(2866-2i\right)t+\left(703-i\right)\right)}{\left(54+2i\right)t+\left(757+i\right)}$$

Variable z cannot be equal to -2.

$$z=\frac{2\left(\left(2866-2i\right)t+\left(703-i\right)\right)}{\left(54+2i\right)t+\left(757+i\right)}\text{, }z\neq -2$$