Exercise 6

Correct: The shape of the level curves indicates that the general form of the function of which these are level curves is equal to $z(x,y) = ax + by + c$ (see Example 1).

On the level curve with value $k=12$ are the points $(x,y)=(0,2)$ and $(x,y)=(6,0)$; hence, we know that
$$(1)~z(0,2) = 2b + c = 12 \iff c = 12-2b \quad \text{and}\quad (2)~z(6,0) = 6a + c = 12\iff c = 12-6a .$$
On the level curve with value $k=18$ are the points $(x,y)=(0,3)$ and $(x,y)=(9,0)$; hence, we know that
$$(3)~z(0,3) = 3b + c = 18\iff c = 18-3b \quad \text{and}\quad (4)~z(9,0) = 9a + c = 18\iff c = 18-9a .$$
Combining $(1)$ and $(3)$ gives
$$\begin{align*} 12-2b &= 18 - 3b\\ b &= 6\\ c &= 12 - 2\cdot6 = 18 - 3\cdot 6 = 0. \end{align*}$$
Combining $(2)$ and $(4)$ gives
$$\begin{align*} 12-6a &= 18 - 9a\\ 3a &= 6\\ a &= 2\\ c &= 12 - 6\cdot2 = 18 - 9\cdot2 = 0. \end{align*}$$
Hence, the function of which the level curves are given is $z(x,y) = 2x + 6y$.

Go on.

Wrong: Pay attention when rewriting.

See Example 1 and try again.

Wrong: This is the equation of the level curve, not of the equation of the function of which the level curve is shown..

See Example 1 and try again.

Wrong: What is the general form of the level curve of a minimum function?

See Example 1 and Example 3.