Why is $a x+b y +c z= d$ a plane and not a line in $\mathbb{R}^3$?

Question

Say for example, i have the bojective function $6x^2 + 12y^2 = z$ shown by the green graph, and i want to minimize the objective function suject to the constraint function $x+y=0.5$ shown by the blue graph

Why does it show that the constraint function(blue) can take on any value of $z$? Shouldn't the blue graph essentially just be a single line with $z=0.5$?

Answer 1

Hint.

You can find the minimum, over the blue intersection curve (parabola)

Answer 2

From your comments I gather that it is the correspondence of equations and geometric lines and surfaces in $\mathbb{R}^3$ that is the problem for you.

Indeed, when working in $\mathbb{R}^2$, there is no distinction between a line (that is, a subset that forms a space of dimension 1) and a hyperplane, which is an (affine) subspace of dimension $\mathbb{R}^{d-1}$, that is, of dimension $1$ if the space is $\mathbb{R}^2$ ($d=2$).

So, let's explain that out.

If you have a space of dimension $d$ (picture it for $d=2$ and $3$, but this is valid in higher dimensions), an equation $$ \sum_{i=1}^d a_i x_i = a_0 $$ allows you to deduce one of the coordinates $x_i$ from the knowledge of all the others. This implies that this equation denotes a subspace whose dimension is just 1 less than the one of the original space: so one equation such as this gives a subspace of dimension $\mathbb{R}^{d-1}$: a line if $d=2$, a plane if $d=3$.

Examples: in 2D, if you set $x_1=x^*_1$ and have $a_1 x_1 + a_2 x_2 = a_0$, then you can deduce what $x_2$ is (unless $a_2 = 0$, then it is a plane parallel to the second axis).

In 3D, if you set $x_1=x_1^*$ (and if $a_1\not=0$) and have $a_1 x_1 + a_2 x_2 + a_3 x_3 = a_0$, then you can deduce only a relationship $a_2 x_2 + a_3 x_3 = (a_0-a_1 x_1^*)$ where the brackets are a constant. You see that you can have many such $x_2$ and $x_3$: in the plane where $x_1$ is the constant $x^*_1$ that you have chosen, $a_2 x_2 + a_3 x_3 = (a_0-a_1 x_1^*)$ denotes a line. All these lines for different $x_1^*$ values form a plane, called an affine hyperplane.

In your figure, $x+y=0.5$ so $a_1=1$, $a_2=1$ and $a_3=0$ is the blue plane you have represented, and those lines are vertical lines $\{ y = 0.5-x^* \text{ and } x=x^*\}$ for different choices of $x^*$.