4) \( \begin{array}{r}-7 x-2 y=-17 \\ -14 x+6 y=16\end{array} \)

To solve the system of equations, let's simplify and manipulate the equations. Start with the first equation and isolate one of the variables: 1. From the first equation: \(-7x - 2y = -17\) Rearranging gives \(2y = -7x + 17\) or \(y = -\frac{7}{2}x + \frac{17}{2}\). 2. Now, use the second equation: \(-14x + 6y = 16\). Substitute \(y\) from the first equation into this equation: \(-14x + 6(-\frac{7}{2}x + \frac{17}{2}) = 16\). Distributing gives: \(-14x - 21x + 51 = 16\). Combine like terms: \(-35x + 51 = 16\). Subtract 51 from both sides: \(-35x = -35\). Divide by \(-35\): \(x = 1\). 3. Now substitute \(x = 1\) back into the equation for \(y\): \(y = -\frac{7}{2}(1) + \frac{17}{2} = -\frac{7}{2} + \frac{17}{2} = \frac{10}{2} = 5\). Thus, the solution to the system of equations is \(x = 1\) and \(y = 5\). --- Now that you have your answer, did you know that systems of equations were used in ancient civilizations? Babylonians were already solving linear equations around 2000 BC! They had methods that surprisingly look similar to our algebraic approaches today. Also, in the real world, solving systems can be incredibly useful! Think about budgeting: if you are balancing incomes and expenses from multiple sources, understanding how each variable impacts your overall budget is a system of equations in action. Mastering this can help keep your financial plans on track!