Percy's error lies in the assumption that any value of \( k \) will lead to no solution for the system of equations. The two equations represent lines in the plane. For the system to have no solution, the lines must be parallel, which occurs when their slopes are equal but their y-intercepts are different.
The first equation, \( 6x + 4y = 14 \), can be rewritten in slope-intercept form as \( y = -\frac{3}{2}x + \frac{7}{2} \) (slope = \(-\frac{3}{2}\)). The second equation, \( 3x + 2y = k \), can be rewritten as \( y = -\frac{3}{2}x + \frac{k}{2} \) (same slope). For the lines to be parallel, they must have the same slope, which they do. However, if \( k \) is chosen such that \( \frac{k}{2} \neq \frac{7}{2} \), the lines will not intersect and thus the system will have no solution.
In summary, only specific values of \( k \) (specifically where \( k \neq 7 \)) lead to no solution, not any real number.