Question Find the solution of the system of equations. \[ \begin{array}{r}-2 x-6 y=8 \\ -6 x-6 y=0\end{array} \] Answer Attempt 1 out of 2

To solve the system of equations, let's start with what we have: 1. \(-2x - 6y = 8\) (Equation 1) 2. \(-6x - 6y = 0\) (Equation 2) From Equation 2, we can simplify it by dividing everything by -6: \[ x + y = 0 \] This implies that: \[ y = -x \] (Equation 3) Now, substitute \(y\) from Equation 3 into Equation 1: \[ -2x - 6(-x) = 8 \] This simplifies to: \[ -2x + 6x = 8 \] \[ 4x = 8 \Rightarrow x = 2 \] Now, plug \(x = 2\) back into Equation 3 to find \(y\): \[ y = -x = -2 \] Thus, the solution to the system of equations is \((2, -2)\). Now, let’s make sure we know the historical significance of why learning systems of equations is vital: throughout history, mathematicians were solving systems to understand everything from trade and commerce to physics and engineering. It’s how they made sense of the world, creating a systematic approach to tackle complex problems! When applying systems of equations in real life, think about any scenario where multiple conditions need to be met – like optimizing production processes in factories. For instance, you might be determining how many of one product to produce against another, taking into account resource limitations and desired outcomes. Understanding and solving systems can help make effective decisions based on available data!