Solve 5i1-i/3-4i-131-4i/5+12i

$$5 i \frac { 1 - i } { 3 - 4 i } - 13 \frac { 1 - 4 i } { 5 + 12 i }$$

Evaluate

$$\frac{202}{65}+\frac{251}{65}i\approx 3.107692308+3.861538462i$$

Solution Steps

Multiply both numerator and denominator of \frac{1-i}{3-4i} by the complex conjugate of the denominator, 3+4i.

$$5i\times \frac{\left(1-i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}-13\times \frac{1-4i}{5+12i}$$

Do the multiplications in \frac{\left(1-i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}.

$$5i\times \frac{7+i}{25}-13\times \frac{1-4i}{5+12i}$$

Divide 7+i by 25 to get \frac{7}{25}+\frac{1}{25}i.

$$5i\left(\frac{7}{25}+\frac{1}{25}i\right)-13\times \frac{1-4i}{5+12i}$$

Multiply 5i and \frac{7}{25}+\frac{1}{25}i to get -\frac{1}{5}+\frac{7}{5}i.

$$-\frac{1}{5}+\frac{7}{5}i-13\times \frac{1-4i}{5+12i}$$

Multiply both numerator and denominator of \frac{1-4i}{5+12i} by the complex conjugate of the denominator, 5-12i.

$$-\frac{1}{5}+\frac{7}{5}i-13\times \frac{\left(1-4i\right)\left(5-12i\right)}{\left(5+12i\right)\left(5-12i\right)}$$

Do the multiplications in \frac{\left(1-4i\right)\left(5-12i\right)}{\left(5+12i\right)\left(5-12i\right)}.

$$-\frac{1}{5}+\frac{7}{5}i-13\times \frac{-43-32i}{169}$$

Divide -43-32i by 169 to get -\frac{43}{169}-\frac{32}{169}i.

$$-\frac{1}{5}+\frac{7}{5}i-13\left(-\frac{43}{169}-\frac{32}{169}i\right)$$

Multiply 13 and -\frac{43}{169}-\frac{32}{169}i to get -\frac{43}{13}-\frac{32}{13}i.

$$-\frac{1}{5}+\frac{7}{5}i-\left(-\frac{43}{13}-\frac{32}{13}i\right)$$

The opposite of -\frac{43}{13}-\frac{32}{13}i is \frac{43}{13}+\frac{32}{13}i.

$$-\frac{1}{5}+\frac{7}{5}i+\left(\frac{43}{13}+\frac{32}{13}i\right)$$

Add -\frac{1}{5}+\frac{7}{5}i and \frac{43}{13}+\frac{32}{13}i to get \frac{202}{65}+\frac{251}{65}i.

$$\frac{202}{65}+\frac{251}{65}i$$

Real Part

$$\frac{202}{65} = 3\frac{7}{65} = 3.1076923076923078$$

Solution Steps

Multiply both numerator and denominator of \frac{1-i}{3-4i} by the complex conjugate of the denominator, 3+4i.

$$Re(5i\times \frac{\left(1-i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}-13\times \frac{1-4i}{5+12i})$$

Do the multiplications in \frac{\left(1-i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}.

$$Re(5i\times \frac{7+i}{25}-13\times \frac{1-4i}{5+12i})$$

Divide 7+i by 25 to get \frac{7}{25}+\frac{1}{25}i.

$$Re(5i\left(\frac{7}{25}+\frac{1}{25}i\right)-13\times \frac{1-4i}{5+12i})$$

Multiply 5i and \frac{7}{25}+\frac{1}{25}i to get -\frac{1}{5}+\frac{7}{5}i.

$$Re(-\frac{1}{5}+\frac{7}{5}i-13\times \frac{1-4i}{5+12i})$$

Multiply both numerator and denominator of \frac{1-4i}{5+12i} by the complex conjugate of the denominator, 5-12i.

$$Re(-\frac{1}{5}+\frac{7}{5}i-13\times \frac{\left(1-4i\right)\left(5-12i\right)}{\left(5+12i\right)\left(5-12i\right)})$$

Do the multiplications in \frac{\left(1-4i\right)\left(5-12i\right)}{\left(5+12i\right)\left(5-12i\right)}.

$$Re(-\frac{1}{5}+\frac{7}{5}i-13\times \frac{-43-32i}{169})$$

Divide -43-32i by 169 to get -\frac{43}{169}-\frac{32}{169}i.

$$Re(-\frac{1}{5}+\frac{7}{5}i-13\left(-\frac{43}{169}-\frac{32}{169}i\right))$$

Multiply 13 and -\frac{43}{169}-\frac{32}{169}i to get -\frac{43}{13}-\frac{32}{13}i.

$$Re(-\frac{1}{5}+\frac{7}{5}i-\left(-\frac{43}{13}-\frac{32}{13}i\right))$$

The opposite of -\frac{43}{13}-\frac{32}{13}i is \frac{43}{13}+\frac{32}{13}i.

$$Re(-\frac{1}{5}+\frac{7}{5}i+\left(\frac{43}{13}+\frac{32}{13}i\right))$$

Add -\frac{1}{5}+\frac{7}{5}i and \frac{43}{13}+\frac{32}{13}i to get \frac{202}{65}+\frac{251}{65}i.

$$Re(\frac{202}{65}+\frac{251}{65}i)$$

The real part of \frac{202}{65}+\frac{251}{65}i is \frac{202}{65}.

$$\frac{202}{65}$$