\( \begin{array}{l}2 x-y-4=0 \\ 3 x+y-9=0\end{array} \) What is the solution set of the given system? \( \{(5,6)\} \) \( \{(6,5)\} \) \( \{(13 / 5,6 / 5)\} \) \( \{(6 / 5,13 / 5)\} \)

To solve the system of equations \(2x - y - 4 = 0\) and \(3x + y - 9 = 0\), we can rearrange both equations to isolate \(y\). The first equation becomes \(y = 2x - 4\) and the second becomes \(y = 9 - 3x\). Setting these equal to each other gives \(2x - 4 = 9 - 3x\). Solving this provides \(5x = 13\) or \(x = \frac{13}{5}\). Substituting \(x\) back into either equation yields \(y = 6\). Therefore, the solution set is \( \{(\frac{13}{5}, 6)\} \). Now, isn’t it exciting how systems of linear equations can intersect at unique points in the vast coordinate plane? For instance, when graphing these lines, they’ll cross perfectly at the coordinate \((\frac{13}{5}, 6)\), showing that math really does create beautiful intersections! If you’re ever stuck solving such systems, remember to check your calculations carefully and look out for signs (plus and minus) that might sneak up on you. It's quite common to misplace a negative sign or mix up addition and subtraction, leading to incorrect results. Double-checking by substituting your found values back into the original equations can save you from these common pitfalls!