Question

The solution of the system of inequalities \(\mathrm{x} \geq 0, \mathrm{y} \geq 0,5 \mathrm{x}+2 \mathrm{y} \geq 10,6 \mathrm{x}+5 \mathrm{y} \leq 30\) is a polygonal region with the vertices (1) \((0,0),(2,0),(0,5),(0,6)\) (2) \((0,0),(5,0),(6,0),(0,2)\) (3) \((2,0),(5,0),(0,6),(0,5)\) (4) \((0,0),(0,5),(6,0),(2,0)\)

Short answer

Question: Identify the polygonal region that is the solution to the system of inequalities: \(x \geq 0\), \(y \geq 0\), \(5x + 2y \geq 10\), \(6x + 5y \leq 30\) Answer: (3) \((2,0), (5,0), (0,6), (0,5)\)

Step-by-step solution

Identify the inequalities

The given system has 4 inequalities: 1. \(x \geq 0\) 2. \(y \geq 0\) 3. \(5x + 2y \geq 10\) 4. \(6x + 5y \leq 30\)

Graph the individual inequalities

We will now graph the inequalities on a graph paper or on a coordinate plane. 1. For \(x \geq 0\), the boundary line is the \(y\)-axis, and the solution area is to the right of the \(y\)-axis. 2. For \(y \geq 0\), the boundary line is the \(x\)-axis, and the solution area is above the \(x\)-axis. 3. For \(5x + 2y \geq 10\), first rewrite it as \(y \geq 5 - \frac{5}{2}x\). The boundary line passes through the points \((2,0)\) and \((0,5)\). The solution area is above this line. 4. For \(6x + 5y \leq 30\), first rewrite it as \(y \leq 6 - \frac{6}{5}x\). The boundary line passes through the points \((5,0)\) and \((0,6)\). The solution area is below this line.

Find the vertices of the solution region

Using the graph of the solution area for all the inequalities, we find the vertices of intersection to be: 1. \((0,0)\) - the intersection of \(x \geq 0\) and \(y \geq 0\) 2. \((2,0)\) - the intersection of \(5x + 2y \geq 10\) and \(y \geq 0\) 3. \((0,5)\) - the intersection of \(5x + 2y \geq 10\) and \(x \geq 0\) 4. \((5,0)\) - the intersection of \(6x + 5y \leq 30\) and \(x \geq 0\) 5. \((0,6)\) - the intersection of \(6x + 5y \leq 30\) and \(y \geq 0\) Comparing the vertices with the given options, we find the correct answer is: (3) \((2,0), (5,0), (0,6), (0,5)\)

Key concepts

Graphical Representation of Inequalities

Graphical representation of inequalities is a visual way to understand the solutions to a set of inequality equations. To graph these inequalities, we use a coordinate plane, which consists of two number lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants. Each inequality can be represented by a region on the plane, delineated by a boundary line.

Where the inequality is 'greater than' or 'greater than or equal to', we shade the region above the boundary line (for y inequalities) or to the right (for x inequalities). Conversely, 'less than' or 'less than or equal to' inequalities are shaded below or to the left. These shaded regions indicate all the possible points (x, y) that satisfy the inequality. When we have a system of inequalities, the graphical solution is typically the overlapping region where all the individual shaded areas meet. This region can take on various shapes, and often, it is a polygon, bounded by the lines that represent the equality part of our inequalities.

Coordinate Plane

The coordinate plane is a critical tool for visualizing and solving mathematical problems involving two dimensions. It is made up of a horizontal axis, known as the x-axis, and a vertical axis, known as the y-axis. These axes intersect at a point called the origin, designated as (0,0). In the context of inequalities, each point on the coordinate plane represents a possible solution to a pair of equations or inequalities.

In a practical exercise, you would plot the boundary line for each equation, also identifying whether the line itself is part of the solution set (dashed or solid line). This demarcation informs you whether to include the line as part of the solution ('equal to') or not. The four quadrants of the plane (I, II, III, and IV), each representing different signs for the x and y coordinates, also help us to quickly identify where the solution regions of certain inequalities will lie.

Polygonal Region Vertices

When you graph a system of linear inequalities, the solution often forms a polygonal region on the coordinate plane. The points, or vertices, of this polygon are where the boundary lines intersect. Each vertex represents an exact solution that satisfies all of the inequalities in the system simultaneously.

To find these points visually, you will want to look for where the lines cross, and analytically you can solve a pair of equations derived from the inequalities. For example, from our exercise, the vertex \( (2,0) \) represents where the line \( 5x + 2y = 10 \) intersects with the x-axis (\(y = 0\)) and so on for each vertex. The region enclosed by connecting these vertices represents all possible solutions to the system of inequalities, and it is crucial to verify that each vertex falls within the shaded region that satisfies all given inequalities.