A system of two linear equations in two variables is a set of two equations where each equation can be written in the form \( ax + by = c \), with \( a \), \( b \), and \( c \) being constants and \( x \) and \( y \) being the variables.
We need to identify which among the given choices fits this description.
Here are the given choices:
1. [tex]\[
\begin{array}{l}
x-2 y+3 z=2 \\
x+2 y-3 z=0
\end{array}
\][/tex]
This is a system of equations, but it has three variables (\(x\), \(y\), and \(z\)), not two.
2. \(4 x+2 y=6\)
This is a single linear equation in two variables, not a system.
3. [tex]\[
\begin{aligned}
x+2 y & =0 \\
5 x-2 y & =12
\end{aligned}
\][/tex]
This is a system of two linear equations in two variables (\(x\) and \(y\)).
4. \(3 x^2 - y = -2\)
This is a single equation and it is not linear since it includes \(x^2\).
5. \(8 x^2 + 4 y = 28\)
This is a single equation and it is not linear since it includes \(x^2\).
6. \(4 x + 2 y = 6\)
This is another single linear equation in two variables.
7. \(x + 2 y = 0\)
This is another single linear equation in two variables.
From these options, the system of two linear equations in two variables is:
[tex]\[
\begin{aligned}
x+2 y & = 0 \\
5 x-2 y & = 12
\end{aligned}
\][/tex]
This system satisfies the requirement of having two linear equations with two variables, \(x\) and \(y\).
Thus, the correct choice is:
[tex]\[
\begin{aligned}
x+2 y & = 0 \\
5 x-2 y & = 12
\end{aligned}
\][/tex]
