Which system is the only inconsistent system? $$ \begin{array}{l} y=x \ y=-x \ y=4 x-3 \ y=4 x+3 \ y=2 x-3 \ 2 y=4 x-6 \ y=x+4 \ y=2 x+4\end{array} $$

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We need to find which pair of equations forms an inconsistent system, meaning the two lines represented by the equations are parallel (identical slopes) but have different y-intercepts.

Consider the pair
$$
\begin{cases}
y = 4x - 3,\[5mm]
y = 4x + 3.
\end{cases}
$$

**Step 1. Compare the slopes.**  
The slope of the first line is $$4$$ (from $$y = 4x-3$$).  
The slope of the second line is also $$4$$ (from $$y = 4x+3$$).  
Since the slopes are the same, the lines are parallel.

**Step 2. Compare the y-intercepts.**  
The first line has a y-intercept of $$-3$$.  
The second line has a y-intercept of $$+3$$.  
Since the y-intercepts are different, the two lines never intersect.

Because parallel lines with different y-intercepts do not intersect, the system
$$
\begin{cases}
y = 4x - 3,\[5mm]
y = 4x + 3
\end{cases}
$$
has no solution and is therefore inconsistent.

Thus, the only inconsistent system is
$$
\begin{cases}
y = 4x - 3,\[5mm]
y = 4x + 3.
\end{cases}
$$
The inconsistent system is: $$ \begin{cases} y = 4x - 3,\ y = 4x + 3. \end{cases} $$

Which system is the only inconsistent system?

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