Q1: What is the solution set of the linear equation $2x_1 - x_2 = 0$ in $\mathbb{R}^2$?

A1: ${ \mathbf{x} \in \mathbb{R}^2 \mid 2 x_1 = x_2 } $

A2: ${ \mathbf{x} \in \mathbb{R}^2 \mid x_1 = 2 x_2 } $

A3: ${ \mathbf{x} \in \mathbb{R}^2 \mid x_1 = 0 } $

A4: ${ \mathbf{x} \in \mathbb{R}^2 \mid x_1 = \frac{1}{2}, x_2 = 1 } $

A5: ${ \mathbf{x} \in \mathbb{R}^2 \mid x_1 = x_2 } $

Q2: What is the solution set for the system $$ \begin{matrix} x_1 + x_2 = 1 \ x_1 - x_2 = 1 \end{matrix} $$

A1: $\left{ \begin{pmatrix} 1 \ 0 \end{pmatrix} \right}$

A2: $\left{ \begin{pmatrix} 0 \ 1 \end{pmatrix} \right}$

A3: ${ \mathbf{x} \in \mathbb{R}^2 \mid x_1 = 1 } $

A4: ${ \mathbf{x} \in \mathbb{R}^2 \mid x_2 = 0 } $

A5: $\emptyset$

Q3: What is the solution set for the system $$ \begin{matrix} x_1 + 2 x_2 = 0 \ x_1 + 2 x_2 = 1 \end{matrix} $$

A1: $\emptyset$

A2: $\left{ \begin{pmatrix} 0 \ 1 \end{pmatrix} \right}$

A3: ${ \mathbf{x} \in \mathbb{R}^2 \mid x_1 = 1 } $

A4: ${ \mathbf{x} \in \mathbb{R}^2 \mid x_2 = 0 } $

A5: $\left{ \begin{pmatrix} \frac{1}{2} \ 1 \end{pmatrix} \right}$