To solve for the translation $T$, we know that the point $(8, -4)$ undergoes two transformations:

1. A translation $T$.
2. A reflection over the line $y = x$, resulting in the final point $(5, -3)$.

Let's break this into two parts:

Step 1: Reflection over the line $y = x$

The reflection of a point $(x, y)$ over the line $y = x$ swaps the coordinates, turning $(x, y)$ into $(y, x)$.

Let the point after translation be $(x', y')$. If $(x', y')$ is reflected over the line $y = x$, the new point will be $(y', x')$, and we know this equals $(5, -3)$. Therefore:

$y' = 5 \quad \text{and} \quad x' = -3$

So, after translation, the point becomes $(-3, 5)$.

Step 2: Finding the translation $T$

Translation $T$ moves the point $(8, -4)$ to $(-3, 5)$. The translation can be expressed as:

$T = (x' - x, y' - y)$

Substitute the coordinates of the original point $(8, -4)$ and the point after translation $(-3, 5)$:

$T = (-3 - 8, 5 - (-4)) = (-11, 9)$

Final Answer:

The translation $T$ is $(-11, 9)$.

Would you like more details or have any questions? Here are 5 related questions to consider:

1. What is the reflection of a point over the line $y = -x$?
2. How do you calculate the translation for a general point $(x, y)$?
3. Can you reflect a point over a vertical or horizontal line instead of $y = x$?
4. How would this problem change if the final point was different?
5. What would happen if you applied two reflections over different lines?

Tip: Always swap the coordinates when reflecting a point over the line $y = x$.