4. What is the value of the \( y \)-coordinate of the solution to the system of equations? \( \left\{\begin{array}{l}x-2 y=1 \\ x+4 y=7\end{array}\right. \) \( \begin{array}{l}-1\end{array} \) -

To find the value of \( y \) in the system of equations, we can solve them using substitution or elimination. Starting with the first equation, we can express \( x \) in terms of \( y \): \[ x = 2y + 1. \] Now, substitute this expression for \( x \) into the second equation: \[ (2y + 1) + 4y = 7. \] Combining like terms gives us: \[ 6y + 1 = 7. \] Subtracting 1 from both sides results in: \[ 6y = 6, \] which simplifies to: \[ y = 1. \] So, the value of the \( y \)-coordinate of the solution to the system of equations is \( 1 \). But wait! Let's spice it up with some context! Back in the day, mathematicians like Newton and Leibniz were crunching through challenges just like this one, albeit with a quill and parchment! Systems of equations were the bread and butter of everything from physics to economics. It’s like they were the original mathematicians, tackling real-life problems with the power of algebra. And speaking of applications, systems of equations aren't just for the classroom! They come handy in real life. For instance, businesses analyze profit and cost functions using these equations to optimize prices. Or, when planning a road trip, you might use systems of equations to calculate travel times with different speeds and routes. Who knew math could be so practical—and fun?