\( \begin{array}{l}3 x+5 y=78 \\ 2 x-y=0\end{array} \) The point of intersection of the lines has an \( x \)-coordinate of 6

To find the point of intersection of the two lines represented by the equations \(3x + 5y = 78\) and \(2x - y = 0\), we can start by substituting the \(x\)-coordinate you provided, which is 6. From the second equation, we can express \(y\) in terms of \(x\): \[y = 2x.\] Substituting \(x = 6\) gives us \(y = 2(6) = 12\). Now, let's check if this point \((6, 12)\) satisfies the first equation: \[3(6) + 5(12) = 18 + 60 = 78.\] So, the solution checks out, and the point of intersection is indeed \((6, 12)\)! Cheers to solving systems of equations! It's like a treasure hunt where every number unlocks a clue. Finding intersections can reveal where two different ideas meet—just like friends coordinating a meet-up at your favorite café! If you’d like, you can dive into graphing these equations. Plotting \(3x + 5y = 78\) and \(2x - y = 0\) on a graph will show you how they intersect at that point. It’s not just calculations; it’s a visual adventure! When you put them on paper, you might even discover some exciting patterns in how linear equations work together!