Solve z=8+i/7-4i

$$z = \frac { 8 + i } { 7 - 4 i }$$

Solve for z

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

Steps for Solving Linear Equation

Multiply both numerator and denominator of \frac{8+i}{7-4i} by the complex conjugate of the denominator, 7+4i.

$$z=\frac{\left(8+i\right)\left(7+4i\right)}{\left(7-4i\right)\left(7+4i\right)}$$

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

$$z=\frac{\left(8+i\right)\left(7+4i\right)}{7^{2}-4^{2}i^{2}}$$

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

$$z=\frac{\left(8+i\right)\left(7+4i\right)}{65}$$

Multiply complex numbers 8+i and 7+4i like you multiply binomials.

$$z=\frac{8\times 7+8\times \left(4i\right)+7i+4i^{2}}{65}$$

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

$$z=\frac{8\times 7+8\times \left(4i\right)+7i+4\left(-1\right)}{65}$$

Do the multiplications in 8\times 7+8\times \left(4i\right)+7i+4\left(-1\right).

$$z=\frac{56+32i+7i-4}{65}$$

Combine the real and imaginary parts in 56+32i+7i-4.

$$z=\frac{56-4+\left(32+7\right)i}{65}$$

Do the additions in 56-4+\left(32+7\right)i.

$$z=\frac{52+39i}{65}$$

Divide 52+39i by 65 to get \frac{4}{5}+\frac{3}{5}i.

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

Assign z

$$z≔\frac{4}{5}+\frac{3}{5}i$$