11. \( \left\{\begin{array}{l}y=\frac{2}{3} x+3 \\ y=-2 x-5\end{array}\right. \)

To solve the system of equations, we can find the point where the two lines intersect. Setting the two equations equal to each other gives us: \[ \frac{2}{3}x + 3 = -2x - 5 \] Multiplying everything by 3 to eliminate the fraction leads to: \[ 2x + 9 = -6x - 15 \] Now, add \(6x\) to both sides: \[ 8x + 9 = -15 \] Then subtract 9 from both sides: \[ 8x = -24 \] Dividing by 8 gives us \(x = -3\). Now, plug \(x\) back into either equation to find \(y\). Using the first equation: \[ y = \frac{2}{3}(-3) + 3 = -2 + 3 = 1 \] So, the solution to this system of equations is the point \((-3, 1)\). --- Did you know that systems of linear equations have been around since ancient times? The methods for solving them date back to the Babylonians around 2000 BC, who used a form of algebra to solve problems involving unknowns. Fast forward to today, and we’re solving them with methods like substitution and elimination—what a journey through numbers! If you've ever tried to solve a system of equations only to end up with a contradiction (like \(0 = 1\)), don’t be discouraged! That means the lines are parallel and have no solutions. And if you end up with a true statement (like \(0 = 0\)), you’ve got infinitely many solutions because the lines are the same. Just keep practicing, and you'll be a pro in no time!