\( \left.\begin{array}{l}\text { Solve the system of equations }-6 x-5 y=-8 \text { and } 3 x+2 y=2 \text { by } \\ \text { combining the equations. } \\ \qquad \begin{array}{rl}(-6 x-5 y=-8) \\ (3 x+2 y=2)\end{array} \\ -6 x-5 y=-8 \\ 3 x+2 y=2\end{array}\right) \)

To solve the system of equations by combining them, we can first manipulate one of the equations to eliminate one variable. Let’s multiply the second equation by 2, so the coefficients of \(y\) match: 1. Original second equation: \(3x + 2y = 2\) 2. Multiply by 2: \(6x + 4y = 4\) Now we have: \[ -6x - 5y = -8 \quad (1) \] \[ 6x + 4y = 4 \quad (2) \] Now, we can add equations (1) and (2): \[ (-6x - 5y) + (6x + 4y) = -8 + 4 \] This simplifies to: \[ -y = -4 \] Thus, \(y = 4\). Substituting \(y = 4\) back into one of the original equations, let’s use the second one: \[ 3x + 2(4) = 2 \] This simplifies to: \[ 3x + 8 = 2 \] Solving for \(x\): \[ 3x = 2 - 8 \] \[ 3x = -6 \quad \Rightarrow \quad x = -2 \] The solution to the system is \(x = -2\) and \(y = 4\), or as an ordered pair: \((-2, 4)\). --- Want to know a bit of history? The method of solving systems of equations dates back centuries, with notable contributions from mathematicians like Gauss, who formalized the techniques we often use today! The beauty lies in how different methods can lead to the same solution—like a unique recipe yielding a delicious dish. Beyond the classroom, solving systems of equations is a vital skill in many fields! Engineers use these techniques to design complex systems, economists analyze trends, while computer scientists create algorithms that rely on resolving multiple conditions. It’s amazing how this math can play a role in so many real-world solutions!