To solve the system of equations, we can substitute the expression for $y$ from the second equation into the first equation. The system is:

$x^2 + y^2 = 153$ $y = -4x$

Substitute $y = -4x$ into the first equation:

$x^2 + (-4x)^2 = 153$

Simplify the equation:

$x^2 + 16x^2 = 153$

Combine like terms:

$17x^2 = 153$

Now, solve for $x^2$ by dividing both sides by 17:

$x^2 = \frac{153}{17}$

$x^2 = 9$

So, the value of $x^2$ is $\boxed{9}$.

Would you like further details or have any other questions?

Here are 5 related questions to consider:

1. What are the possible values of $x$ given $x^2 = 9$?
2. How would you find the corresponding $y$ values for each $x$?
3. What is the distance between the origin and the point $(x, y)$?
4. Can the method of substitution be used in all systems of equations?
5. How would the solution change if the constant in the first equation were different?

Tip: When dealing with systems of equations, substitution is a powerful method, especially when one equation is already solved for a variable.