We define \begin{equation*}\mathbb{L}(a,b,c):=\left \{\begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2\mid ax+by=c\right \}\end{equation*}
Determine $\mathbb{L}(1,1,0)$ and $\mathbb{L}(1,1,5)$ and give the the geometric description of these sets.
Determine $\mathbb{L}(0,1,3)$ and $\mathbb{L}(2,0,1)$ and give the the geometric description of these sets.
I have done the following:
- $\displaystyle{\mathbb{L}(1,1,0)=\left \{\begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2\mid x+y=0\right \}=\left \{\begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2\mid y=-x\right \}}$
It is the set of the points of the line $y=-x$.
$\displaystyle{\mathbb{L}(1,1,5)=\left \{\begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2\mid x+y=5\right \}=\left \{\begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2\mid y=5-x\right \}}$
It is the set of the points of the line $y=5-x$.
- $\displaystyle{\mathbb{L}(0,1,3)=\left \{\begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2\mid y=3\right \}}$
It is the set of the points of the line $y=3$.
$\displaystyle{\mathbb{L}(2,0,1)=\left \{\begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2\mid 2x=1\right \}=\left \{\begin{pmatrix}x\\ y\end{pmatrix}\in \mathbb{R}^2\mid x=\frac{1}{2}\right \}}$
It is the set of the points of the line $x=\frac{1}{2}$.
Is everything correct and complete?