ModulusWhat is |z| if i - iz = 2 ?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the following formula to evaluate the absolute value of `z = x + i*y` , such that:

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

You need to find x and y, hence, substituting `x + i*y` in the equation provided by the problem, yields:

`i - i(x + i*y) = 2`

You need to perform the multiplications such that:

`i - x*i - y*i^2 = 2`

You need to substitute -1 for `i^2` such that:

`i - x*i + y = 2`

You need to factor out i such that:

`i(1 - x) + y = 2`

Equating the real parts and imaginary parts both sides yields:

`{(1 - x = 0),(y = 2):} => {(x = 1),(y = 2):}`

You may evaluate the absolute value of  such that:

`|z| = sqrt(1^2 + 2^2) => |z| = sqrt5`

Hence, evaluating the absolute value of z, under the given conditions, yields `|z| = sqrt5.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll re-write z, isolating z to the left side. For this reason, we'll subtract i both sides:

i - iz = 2

-iz = 2 - i

We'll divide by -i:

z = (2-i)/-i

Since we have to put z in the rectangular form:

z = x + i*y, we'll multiply the ratio by the conjugate of -i, that is i.

z = i*(2-i)/-i^2

But i^2 = -1

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

We'll remove the brackets:

z = 2i - i^2

z = 1 + 2i

gThe modulus of z: |z| = sqrt (x^2 + y^2)

We'll identify x = 1 and y = 2.

|z| = sqrt(1 + 4)

|z|  =sqrt 5

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