Solve (3+4i)z=7+i

$$( 3 + 4 i ) z = 7 + i$$

Solve for z

$$z=1-i$$

Steps for Solving Linear Equation

Divide both sides by 3+4i.

$$z=\frac{7+i}{3+4i}$$

Multiply both numerator and denominator of \frac{7+i}{3+4i} by the complex conjugate of the denominator, 3-4i.

$$z=\frac{\left(7+i\right)\left(3-4i\right)}{\left(3+4i\right)\left(3-4i\right)}$$

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

$$z=\frac{\left(7+i\right)\left(3-4i\right)}{3^{2}-4^{2}i^{2}}$$

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

$$z=\frac{\left(7+i\right)\left(3-4i\right)}{25}$$

Multiply complex numbers 7+i and 3-4i like you multiply binomials.

$$z=\frac{7\times 3+7\times \left(-4i\right)+3i-4i^{2}}{25}$$

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

$$z=\frac{7\times 3+7\times \left(-4i\right)+3i-4\left(-1\right)}{25}$$

Do the multiplications in 7\times 3+7\times \left(-4i\right)+3i-4\left(-1\right).

$$z=\frac{21-28i+3i+4}{25}$$

Combine the real and imaginary parts in 21-28i+3i+4.

$$z=\frac{21+4+\left(-28+3\right)i}{25}$$

Do the additions in 21+4+\left(-28+3\right)i.

$$z=\frac{25-25i}{25}$$

Divide 25-25i by 25 to get 1-i.

$$z=1-i$$