Solve 2+(5-i)z-14-7i=0

$$2 + ( 5 - i ) z - 14 - 7 i = 0$$

Solve for z

$$z=\frac{53}{26}+\frac{47}{26}i\approx 2.038461538+1.807692308i$$

Steps for Solving Linear Equation

Combine the real and imaginary parts in 2-14-7i.

$$\left(5-i\right)z+2-14-7i=0$$

Add 2 to -14.

$$\left(5-i\right)z-12-7i=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$\left(5-i\right)z-7i=12$$

Add 7i to both sides.

$$\left(5-i\right)z=12+7i$$

Divide both sides by 5-i.

$$z=\frac{12+7i}{5-i}$$

Multiply both numerator and denominator of \frac{12+7i}{5-i} by the complex conjugate of the denominator, 5+i.

$$z=\frac{\left(12+7i\right)\left(5+i\right)}{\left(5-i\right)\left(5+i\right)}$$

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

$$z=\frac{\left(12+7i\right)\left(5+i\right)}{5^{2}-i^{2}}$$

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

$$z=\frac{\left(12+7i\right)\left(5+i\right)}{26}$$

Multiply complex numbers 12+7i and 5+i like you multiply binomials.

$$z=\frac{12\times 5+12i+7i\times 5+7i^{2}}{26}$$

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

$$z=\frac{12\times 5+12i+7i\times 5+7\left(-1\right)}{26}$$

Do the multiplications in 12\times 5+12i+7i\times 5+7\left(-1\right).

$$z=\frac{60+12i+35i-7}{26}$$

Combine the real and imaginary parts in 60+12i+35i-7.

$$z=\frac{60-7+\left(12+35\right)i}{26}$$

Do the additions in 60-7+\left(12+35\right)i.

$$z=\frac{53+47i}{26}$$

Divide 53+47i by 26 to get \frac{53}{26}+\frac{47}{26}i.

$$z=\frac{53}{26}+\frac{47}{26}i$$