To solve this problem, we analyze the transformations step by step:

Step 1: Rotation of the line $y = x$ by $90^\circ$ counterclockwise

When a line is rotated about the origin, its slope changes. The line $y = x$ has a slope of 1. A $90^\circ$ counterclockwise rotation transforms the slope $m$ to $-1/m$. Thus, the new slope becomes: $m = -1/1 = -1$ This gives us the equation of the rotated line: $y = -x$

Step 2: Translating the line $y = -x$ left 5 units

Translation to the left affects the $x$-coordinate by $-5$. To translate, replace $x$ in the equation $y = -x$ with $(x + 5)$. Substituting: $y = -(x + 5)$ Simplify: $y = -x - 5$

Thus, the equation of the resulting line is: $\boxed{y = -x - 5}$

Let me know if you'd like further clarification or details! Below are some related questions to deepen understanding:

1. What happens to a line if it's rotated $180^\circ$ about the origin instead of $90^\circ$?
2. How does translating a line vertically differ from translating it horizontally?
3. If the line $y = 2x$ is rotated $90^\circ$ counterclockwise, what is the resulting equation?
4. Can a translation alone ever change the slope of a line? Why or why not?
5. What is the general process for rotating a line about a point other than the origin?

Tip: When rotating a line, focus on how the slope changes according to the rotation angle. For $90^\circ$, reciprocal and sign changes are key!