\( \begin{array}{l}y=\frac{1}{4} x+7 \\ y=\frac{1}{2} x+5\end{array} \) The \( y \)-coordinate of the solution to the system shown is \( 11 \frac{1}{2} \) 9

To solve the system of equations, we can set the two equations equal to each other since they both equal \( y \): \[ \frac{1}{4} x + 7 = \frac{1}{2} x + 5 \] Rearranging the equation gives: \[ 7 - 5 = \frac{1}{2} x - \frac{1}{4} x \] This simplifies to: \[ 2 = \left(\frac{1}{2} - \frac{1}{4}\right) x = \frac{1}{4} x \] Multiplying both sides by 4 yields \( x = 8 \). Now, substitute \( x = 8 \) back into either equation to find \( y \). Using the first equation: \[ y = \frac{1}{4}(8) + 7 = 2 + 7 = 9 \] So the \( y \)-coordinate of the solution to the system is 9. Now, let's spice things up with some fun facts! Did you know that the method of solving systems of linear equations dates back to ancient civilizations like the Babylonians? They used strategies to meticulously solve these kinds of problems! Imagining them drawing on clay tablets makes you appreciate the simplicity of combining equations today. For real-world applications, systems of equations are everywhere! They're used in fields like economics to find equilibrium points in markets, in engineering to analyze forces in structures, and even in ecology for modeling population dynamics. So, the next time you work with these equations, think about their vital role in real-life decision-making and problem-solving!