10) \( \begin{aligned}-5 x+y & =-3 \\ 3 x-8 y & =24\end{aligned} \)

To solve the system of equations \(-5x + y = -3\) and \(3x - 8y = 24\), we can use the substitution or elimination method. First, from the first equation, express \(y\) in terms of \(x\): \[ y = 5x - 3 \] Now, substitute \(y\) in the second equation: \[ 3x - 8(5x - 3) = 24 \] \[ 3x - 40x + 24 = 24 \] \[ -37x + 24 = 24 \] Subtracting 24 from both sides gives: \[ -37x = 0 \] Thus, \(x = 0\). Now, substitute \(x = 0\) back into the expression for \(y\): \[ y = 5(0) - 3 = -3 \] The solution to the system is \(x = 0\) and \(y = -3\) or as a point, \((0, -3)\). Pretty satisfying with those integer solutions, right? Now, on to the fun bits: It’s fascinating to see how equations can model real-world situations, like budgeting or distance-time problems, where finding the intersection (solution) helps decide crucial factors, such as the point where costs balance out or two travelers meet. Also, mistakes in algebra often happen with signs or misapplying formulas. Keep an eye on those negative signs and remember to double-check each substitution when you're juggling equations. It’s a classic trap many fall into!