Question 17 Pause Zoom Asystem of equations is shown below. \[ \left\{\begin{array}{l} 8 x-2 y=-4 \\ 4 x-y=-2 \end{array}\right. \] What is the solution to this system of equations? A. There is no solution to this system of equations. B. There are an infinite number of solutions to this system of equations. C. \( (0,-2) \) D. \( (2,0) \)

Let's start by solving the system of equations step by step. First, we can use the second equation \( 4x - y = -2 \) to express \( y \) in terms of \( x \): \[ y = 4x + 2 \] Next, we substitute this expression for \( y \) into the first equation \( 8x - 2y = -4 \): \[ 8x - 2(4x + 2) = -4 \] Simplifying gives: \[ 8x - 8x - 4 = -4 \] This simplifies to \( -4 = -4\), which is true for any value of \( x \). Thus, this indicates that the equations are dependent and describe the same line, leading to an infinite number of solutions for the system. So the answer is **B. There are an infinite number of solutions to this system of equations.** Want a little history on systems of equations? They date back to ancient civilizations! The Babylonians were solving linear equations as early as 2000 BCE, paving the way for future advancements in algebra. Fast forward to now, and we use systems of equations in everything from economics to engineering! For some real-world application, consider how traffic systems use equations to optimize flow. By creating models based on traffic patterns and predictive analytics, engineers can determine the best solutions for managing traffic lights, reducing congestion, and improving overall travel times. Systems of equations help solve these complex problems!