Animated Solution for Mathematics - Complex Numbers: Let $z$ be a complex number such that $|z|+z=3+i$ (where $i=\sqrt{-1}$). Then $|z|$ is equal to:

Welcome, fellow traveler of the mathematical landscape. Today, we are going to unravel a problem that, at first glance, might seem like a simple algebraic puzzle, but it is actually a beautiful dance between geometry and algebra.

We are given the equation $|z|+z=3+i$. Before we dive into the mechanics, visualize this in the complex plane where $z$ is a vector.

The term $|z|$ is a scalar—a magnitude. We are adding a scalar to the vector itself, and the result is the point $(3,1)$. This tells us that the real part of our complex number $z$ is being "stretched" by its own magnitude.

To solve this, we assume the standard form of a complex number: $z=x+iy$, where $x$ and $y$ are real numbers. The magnitude is defined as $|z|=\sqrt{x^2+y^2}$.

Substituting these into our original equation, we get:

For these two complex numbers to be equal, their real and imaginary parts must match independently. This yields two distinct equations:

1. Imaginary part: $y=1$

2. Real part: $\sqrt{x^2+y^2}+x=3$

Since $y=1$, we substitute this into our second equation to obtain:

To solve for $x$, we isolate the radical:

We must note the constraint $3-x>0$, or $x<3$, because the square root function is always non-negative. Squaring both sides, we get:

Expanding the right side yields:

The $x^2$ terms cancel out perfectly, leaving us with $1=9-6x$. Solving this linear equation:

We have found $x=\frac{4}{3}$ and $y=1$. To find the magnitude $|z|$, we look back at our real part equation: $|z|+x=3$.

This implies:

Substituting our value for $x$:

The logic holds, the math is clean, and the result is beautiful. You have successfully navigated the trap of extraneous roots and arrived at the correct magnitude of $|z|=\frac{5}{3}$.