Find the points of intersection between the curve xy = 12 and the line y = 7 − x.

Given the curve $xy = 12$ and the line $y = 7 - x$.
Substitute $y = 7 - x$ into the curve equation:

$$x(7 - x) = 12$$

$$7x - x^2 = 12$$

Move all terms to one side to get a standard quadratic equation:

$$x^2 - 7x + 12 = 0$$

Factor the quadratic equation:
We need two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4.

$$(x - 3)(x - 4) = 0$$

This gives two possible values for $x$:

$$x - 3 = 0 \implies x_1 = 3$$ $$x - 4 = 0 \implies x_2 = 4$$

For $x_1 = 3$:

$$y_1 = 7 - 3$$ $$y_1 = 4$$

The first point of intersection is $(3, 4)$.

For $x_2 = 4$:

$$y_2 = 7 - 4$$ $$y_2 = 3$$

The second point of intersection is $(4, 3)$.

The coordinates of the two points are $(3, 4) \text{ and } (4, 3)$.