# Find the absolute value of z if 3z+i = 2z -i + 5

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We are given 3z + i = 2z - i + 5. We have to find the absolute value of z.

First let us find z.

3z + i = 2z - i + 5

move all the terms with z to one side and the other terms to the opposite side

=> 3z - 2z = -i - i +5

add and subtract similar terms

=> z = -2i + 5

So we get z = -2i + 5

Now the absolute value of z = a + ib is sqrt (a^2 + b^2)

Therefore the absolute value of z is sqrt ((-2)^2 + 5^2)

=sqrt ( 4 + 25)

=sqrt 29.

**The required absolute value is sqrt 29.**

Given the complex number 3z+i = 2z -i + 5.

We need to find the absolute values for z.

First, we need to rewrite z as a complex number into the form z= a+ bi.

==> 3z + i = 2z - i + 5

We will combine like terms.

==> 3z - 2z + i + i - 5 = 0

==> z +2i - 5 = 0

Now we will isolate z on the left side.

==> z = 5 - 2i

Then , a= 5 and b = -2

Then the absolute value for z is given by:

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

==> l z l = sqrt( 5^2 + -2^2)

= sqrt( 25+ 4) = sqrt29.

**Then, the absolute value for z is l z l = sqrt29.**

To find the absolute value of z in 3z+i= 2z-i+5.

The absolute value of z = x+yi is |z| = |x+yi| = sqrt{x^2+y^2}.

We first solve for z in 3z+i= 2z-i+5.

3z-2z = -i+5-i.

z = 5-2i.

Therefore |z| = |5-2i|.

|z| = sqrt{5^2+(-2)^2}.

|z| = sqrt{25+4) = sqrt29.

|z| = sqrt 29.

To determine the absolute value of z, we'll have to put z into the rectangular form:

z = x + i*y

|z| = sqrt(x^2 + y^2)

To determine the absolute value, we'll have to isolate z to the left side:

3z+i = 2z -i + 5

We'll subtract 2z:

3z - 2z = -i - i + 5

z = 5 - 2i

We'll determine the absolute value:

|z| = sqrt(5^2 + (-2)^2)

|z| = sqrt (25+4)

**|z| = sqrt 29**

**The absolute value of the complex number is ****|z| = sqrt 29.**