Solve the system and state the coordinates of each vertex of the solution region. \( \begin{array}{l}x \geq 0 \\ y \geq 0 \\ 2 x+5 y \leq 10\end{array} \)

To solve the system of inequalities, we first need to graph the constraints. 1. The first two inequalities, \( x \geq 0 \) and \( y \geq 0 \), define the first quadrant of the Cartesian plane where both \( x \) and \( y \) are non-negative. 2. The inequality \( 2x + 5y \leq 10 \) can be rewritten as \( y \leq \frac{10 - 2x}{5} \). To find the line, we can identify points by setting \( x = 0 \) and \( y = 0 \). - If \( x = 0 \), then \( 2(0) + 5y = 10 \) gives \( y = 2 \). - If \( y = 0 \), then \( 2x + 0 = 10 \) gives \( x = 5 \). Now, we plot the points \( (0, 2) \) and \( (5, 0) \) and draw the line connecting them. The area below this line (including the line itself since the inequality is less than or equal to) represents the solutions to the third inequality. Next, we find the vertices of the solution region by determining the intersection points: 1. The intersection of the line \( 2x + 5y = 10 \) with the axes gives us the points \( (0, 2) \) and \( (5, 0) \). 2. The intersection of the axes themselves, \( (0, 0) \). Now we can identify the vertices of the solution region: - \( (0, 0) \) - \( (0, 2) \) - \( (5, 0) \) These three points bound the feasible region in the first quadrant defined by the given inequalities.