To classify the given system of linear equations:
[tex]\[
\left\{\begin{array}{l}
x - 2y = 4 \\
y = \frac{1}{2}x + 6
\end{array}\right.
\][/tex]
Let's go through a detailed, step-by-step process.
1. Rewrite the second equation in the standard form:
The first equation is already in the standard form:
[tex]\[
x - 2y = 4
\][/tex]
The second equation given is:
[tex]\[
y = \frac{1}{2}x + 6
\][/tex]
Multiply both sides of the second equation by 2 to clear the fraction:
[tex]\[
2y = x + 12
\][/tex]
Now, let's rewrite it in the standard form:
[tex]\[
x - 2y = -12
\][/tex]
2. Compare the two equations:
Now we have:
[tex]\[
\left\{\begin{array}{l}
x - 2y = 4 \\
x - 2y = -12
\end{array}\right.
\][/tex]
We can see that the left-hand sides of both equations are identical, but the right-hand sides are different.
3. Analyze the equations:
Both equations have the same left-hand side, [tex]\( x - 2y \)[/tex].
This means they are parallel lines with no points in common, as their right-hand sides are different (4 and -12).
Consequently, the system of equations has no solution because parallel lines do not intersect.
Such a system is classified as inconsistent.
Thus, the classification that describes the system of linear equations is:
[tex]\[
\boxed{\text{inconsistent}}
\][/tex]
