# What is the absolute value of z if 3z -9i = 8i + z + 4.

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3z = 9i = 8i + z + 4

We need to determine the absolute value of.

First let us determine z as a form of the complex number z = a + bi

First we will combine like terms.

==> 3z - z = 8i - 9i + 4

==> 2z = 4 - i

Now we will divide by 2.

==> z = (4-i) /2

==> z = 2 - (1/2) i

Now that we wrote z into the standard form, we will calculate the absolute value.

We know that:

l z l = sqrt( a^2 + b^2).

==> l z l = sqrt( 2^2 + (1/2)^2

= sqrt( 4 + 1/4)

= sqrt(17/4) = sqrt17 / 2

**==> l z l = sqrt17 / 2**

**Then, the absolute value of z is sqrt17 / 2**

The absolute value of z is the distance:

|z| = sqrt(Re(z)^2 + Im(z)^2)

We'll have to determine the rectangular form of z, from the given expression:

3z -9i = 8i + z + 4

We'll isolate z to the left side. For this reason, we'll subtract z both sideS;

2z - 9i = 8i + 4

We'll add 9i both sides:

2z = 4 + 17i

We'll divide by 2:

z = 2 + 17i/2

Now, we can identify the real part and the imaginary part of the complex number, in order to compute the distance |z|.

Re(z) = 2

Im(z) = 17/2

|z| = sqrt(4 + 289/4)

**|z| = sqrt305/2**

What is the absolute value of z if 3z -9i = 8i + z + 4.

We first solve for z from the equation:

3z-9i = 8i+z+4.

Add 9i-3z to both sides and we get:

3-z = 8i+9i+4

2z = 17i+4.

z = (4+17i)/2

z = 2+(17/2)i

We know that absolut value of z = |z| = x+yi = sqrt(x^2+y^2).

Therrefore |z} = sqrt{(2)^2+(17/2)^2}.

|z| = sqrt{16+289/4}.

|z| = sqrt{(256+289)/4}.

|z| = (1/2){sqr545).