Answer
No solution (inconsistent system)
Work Step by Step
Given: $$\begin{cases}
y &= -2x+8\\
x+\frac{1}{2}y &=3.
\end{cases}$$ First, rewrite the second equation in slope intercept form.
$$\begin{aligned}
x+\frac{1}{2}y &=3 \\
\frac{1}{2}y &= -x+3\\
2\cdot\left(\frac{1}{2}y \right)&= 2\cdot\left(-x+3\right)\\
y&=-2x+6.
\end{aligned}$$ Compare the two equations $$\begin{cases}
y &= -2x+8\\
y &=-2x+6.
\end{cases}$$ By comparison, we see that both lines have the same slopes but different y-intercept. This means that the lines are parallel and therefore inconsistent. There is no solution for the system. Tom's solution is the correct answer. However, the error in Matt's solution is seen here. He started by substituting for $y$ from the first equation into the second. $$\begin{aligned}
x+\frac{1}{2}\left(-2x+8\right) &= 3\\
x-x+4&= 3\\
4&=3
\end{aligned}$$ which is clearly false. He also claimed that the system is consistent with dependent lines with infinitely many solutions. All of these claims are clearly false.
