2. \( \left\{\begin{array}{c}3 x+5 y=31 \\ x=4 y-1\end{array}\right. \)

To solve this set of equations, start with the second equation \( x = 4y - 1 \). Substitute this expression for \( x \) into the first equation: \[ 3(4y - 1) + 5y = 31. \] This simplifies to \( 12y - 3 + 5y = 31 \), or \( 17y - 3 = 31 \). Now, add 3 to both sides: \( 17y = 34 \). Dividing both sides by 17 gives \( y = 2 \). Next, substitute \( y \) back into the equation \( x = 4y - 1 \): \[ x = 4(2) - 1 = 8 - 1 = 7. \] So the solution to the system is \( x = 7 \), \( y = 2 \). These equations can represent various scenarios, such as determining the quantities of two products given their constraints (e.g., budget or availability). For example, if \( x \) represents the number of bicycles and \( y \) represents the number of helmets produced within a specific budget, solving these equations can help manufacturers optimize their production to meet demands. A common mistake when solving systems of equations is forgetting to distribute terms properly. In this case, while substituting \( x \) values into the first equation, ensure that you're accurately applying the distributive property's multiplication across every term—missteps here can lead to incorrect solutions, so double-check your calculations!