Solve z=(a+5)i^6+(a-3)i^7

$$z = ( a + 5 ) i ^ { 6 } + ( a - 3 ) i ^ { 7 }$$

Solve for a

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

Steps for Solving Linear Equation

Calculate i to the power of 6 and get -1.

$$z=\left(a+5\right)\left(-1\right)+\left(a-3\right)i^{7}$$

Use the distributive property to multiply a+5 by -1.

$$z=-a-5+\left(a-3\right)i^{7}$$

Calculate i to the power of 7 and get -i.

$$z=-a-5+\left(a-3\right)\left(-i\right)$$

Use the distributive property to multiply a-3 by -i.

$$z=-a-5-ia+3i$$

Combine -a and -ia to get \left(-1-i\right)a.

$$z=\left(-1-i\right)a-5+3i$$

Swap sides so that all variable terms are on the left hand side.

$$\left(-1-i\right)a-5+3i=z$$

Add 5 to both sides.

$$\left(-1-i\right)a+3i=z+5$$

Subtract 3i from both sides.

$$\left(-1-i\right)a=z+5-3i$$

The equation is in standard form.

$$\left(-1-i\right)a=z+\left(5-3i\right)$$

Divide both sides by -1-i.

$$\frac{\left(-1-i\right)a}{-1-i}=\frac{z+\left(5-3i\right)}{-1-i}$$

Dividing by -1-i undoes the multiplication by -1-i.

$$a=\frac{z+\left(5-3i\right)}{-1-i}$$

Divide z+\left(5-3i\right) by -1-i.

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

Solve for z

$$z=\left(-1-i\right)a+\left(-5+3i\right)$$