4. \( \begin{array}{l}y=-3 x-9 \\ 5 x+2 y=-16\end{array} \)
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To solve the system of equations, let’s start with the first equation: \(y = -3x - 9\). We can substitute this expression for \(y\) into the second equation, \(5x + 2y = -16\). Substituting gives us: \[ 5x + 2(-3x - 9) = -16. \] Simplifying this, we have: \[ 5x - 6x - 18 = -16, \] which leads us to: \[ -x - 18 = -16. \] Adding 18 to both sides results in: \[ -x = 2, \] thus, \[ x = -2. \] Now, we can substitute \(x = -2\) back into the first equation to find \(y\): \[ y = -3(-2) - 9 = 6 - 9 = -3. \] The solution to the system is \( (x, y) = (-2, -3) \). Would you like to explore more about the theoretical implications of this solution or delve into some practical examples where such systems are used?