Question

which system is the only inconsistent system?
○ $2x = y$ $x = y$
○ $2x + y = 8$ $2x - y = 8$
○ $2x + y = 8$ $2x + y = 10$
○ $2x + y = 8$ $4x + 2y = 16$

Explanation:

To determine an inconsistent system, we check if the two equations represent parallel lines (no solution). An inconsistent system has no solution, which occurs when the lines have the same slope but different y - intercepts.

Step 1: Analyze the first system: \(

$$\begin{cases}2x = y\\x = y\end{cases}$$

\)
Rewrite the equations in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept).
The first equation \(2x=y\) can be written as \(y = 2x\) (slope \(m_1 = 2\), \(b_1=0\)).
The second equation \(x = y\) can be written as \(y=x\) (slope \(m_2 = 1\), \(b_2 = 0\)).
Since the slopes are different, the lines intersect at a point. So, this system is consistent.

Step 2: Analyze the second system: \(

$$\begin{cases}2x + y=8\\2x - y=8\end{cases}$$

\)
Rewrite the equations in slope - intercept form.
For \(2x + y=8\), we get \(y=- 2x + 8\) (slope \(m_1=-2\), \(b_1 = 8\)).
For \(2x - y=8\), we can rewrite it as \(y=2x - 8\) (slope \(m_2 = 2\), \(b_2=-8\)).
Since the slopes are different, the lines intersect at a point. So, this system is consistent.

Step 3: Analyze the third system: \(

$$\begin{cases}2x + y=8\\2x + y=10\end{cases}$$

\)
Rewrite the equations in slope - intercept form.
For \(2x + y=8\), we have \(y=-2x + 8\) (slope \(m_1=-2\), \(b_1 = 8\)).
For \(2x + y=10\), we have \(y=-2x + 10\) (slope \(m_2=-2\), \(b_2 = 10\)).
The slopes are the same (\(m_1=m_2=-2\)) and the y - intercepts are different (\(b_1
eq b_2\)). So, the lines are parallel and do not intersect. This system is inconsistent.

Step 4: Analyze the fourth system: \(

$$\begin{cases}2x + y=8\\4x + 2y=16\end{cases}$$

\)
Simplify the second equation. Divide \(4x + 2y=16\) by 2, we get \(2x + y=8\), which is the same as the first equation. So, the two equations represent the same line. There are infinitely many solutions, and the system is consistent (dependent).

Answer:

The system \( \boldsymbol{

$$\begin{cases}2x + y=8\\2x + y=10\end{cases}$$

} \) (the third option: \(2x + y = 8\); \(2x + y = 10\)) is the only inconsistent system.