Part B \( \begin{array}{l}\text { b. } y=2 x+5 \\ x+2 y=-5\end{array} \) A) one solution B) infinitely many solutions \( \square \) D) no solution \( \square \) E) inconsistent \( \square \) F) independent \( \square \) G) dependent

To determine the number of solutions to the system of equations, let's rewrite them in a clearer form: 1. \( y=2x+5 \) (Equation 1) 2. Rearranging \( x + 2y = -5 \) gives us \( y = \frac{-x-5}{2} \) (Equation 2). Next, we observe both equations. The first equation has a slope of 2, while the rearranged second equation has a slope of -0.5. Since their slopes are different, the lines intersect at exactly one point. Thus, this system of equations has one unique solution, making it consistent and independent. Now let's expand your knowledge about this topic! This type of linear equation system dates back to ancient civilizations, where mathematicians like Babylonian scholars used similar principles to solve problems involving land distribution and trade. It’s fascinating to think how these foundational concepts laid the groundwork for modern algebra! In real life, these linear equations can represent various scenarios, such as budgeting or predicting profits. Imagine you're planning a party and trying to figure out the budget based on the number of guests; these equations can help you visualize and determine your limits effectively.