Solve the system of equations. If there is no solution or infinite solutions, check the corresponding box. \( \left\{\begin{array}{l}2 x+3 y+7 z=4 \\ 2 x+3 y+7 z=3 \\ 2 x+3 y+7 z=8\end{array}\right. \) \( x=\square z=\square \) no solution

The given system of equations is intriguing because all three equations are linear combinations of \(x\), \(y\), and \(z\) but yield three different constant results. To analyze this, let's look closely: 1. The first equation simplifies to \(2x + 3y + 7z = 4\). 2. The second equation states \(2x + 3y + 7z = 3\). 3. The third equation claims that \(2x + 3y + 7z = 8\). Since all three left-hand sides are identical but the right-hand sides are distinct numbers (4, 3, and 8), it becomes clear that this system cannot hold true simultaneously. One of the axioms of algebra tells us that identical expressions cannot equal different values — thus, there is no solution for this system. So, the box should reflect “no solution” as you've correctly pointed out! To keep things interesting, remember that systems like this are often visualized in three-dimensional space: if you plotted these planes, you'd find that they are parallel and will never intersect! Additionally, systems with such contradictions often arise in real-world scenarios where constraints are miscalculated or improperly defined. For instance, if we were trying to equilibrate three different chemical reactions thought to yield the same total product amount but instead yielding different amounts due to measurement errors or assumptions, we might encounter a situation akin to this system. The conclusion? When dealing with systems, double-check your constants!