Problem 117
Question
Will help you prepare for the material covered in the next section. a. Graph the solution set of the system:$$\left\\{\begin{array}{c}x \geq 0 \\\y \geq 0 \\\3 x-2 x \leq 6 \\\y \leq-x+7\end{array}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(2 x+5 y\) at each of the points obtained in part (b)
Step-by-Step Solution
Verified Answer
From the graph, the corner points are (0,0), (0,7) and (2,5). Evaluating \(2x+5y\) at these points results in 0, 35 and 20 respectively.
1Step 1: Graph the inequalities
Graph each of the inequalities \(x \geq 0\), \(y \geq 0\), \(3 x-2 y \leq 6\), and \(y \leq-x+7\) on the same plane. Identification of the shared region that satisfies all the inequalities is crucial.
2Step 2: Find the corner points
The corners of the graphed region are the points of intersection of the inequalities. There are three ways these lines can intersect: the y-axis (\(x=0\)), the x-axis (\(y=0\)), and where \(3x-2y=6\) intersects \(y=-x+7\). Solve these points of intersection.
3Step 3: Evaluate at the corner points
Evaluate the expression \(2x+5y\) at each of the corner points obtained in step 2. This step involves direct substitution of x and y values of the corner points into the expression.
Key Concepts
Systems of InequalitiesCorner Points of PolygonsInequality GraphingLinear Programming
Systems of Inequalities
Understanding systems of inequalities is fundamental when we deal with multiple constraints that need to be satisfied simultaneously. In contrast to single equations, a system of inequalities represents several conditions that an answer must fulfill. Take, for example, the basic inequalities in our exercise:
\(x \geq 0\), \(y \geq 0\), \(3x - 2y \leq 6\), and \(y \leq -x + 7\). Each of these inequalities defines a half-plane on a graph, and the solution to the system is the region where these half-planes overlap.
When graphing a system of inequalities, we're seeking the entire set of points that satisfy all inequalities at once. This region is visually represented as the shared space where all the shaded areas from each inequality intersect on a graph.
\(x \geq 0\), \(y \geq 0\), \(3x - 2y \leq 6\), and \(y \leq -x + 7\). Each of these inequalities defines a half-plane on a graph, and the solution to the system is the region where these half-planes overlap.
When graphing a system of inequalities, we're seeking the entire set of points that satisfy all inequalities at once. This region is visually represented as the shared space where all the shaded areas from each inequality intersect on a graph.
Corner Points of Polygons
Identifying the corner points of polygons is crucial when analyzing graphs of inequalities. These corner points, also known as vertices, are where the boundaries of the inequalities intersect. In our exercise, these boundaries are the lines represented by the equations \(x=0\), \(y=0\), \(3x - 2y = 6\), and \(y = -x + 7\).
Where these lines intersect determines the shape of the feasible region, the area where all the inequalities in the system hold true. The corner points are informative because in many optimization problems, such as those in linear programming, the optimal solution lies at one of these vertices. To find these points methodically, one has to calculate the points of intersection between all possible pairs of lines making up the inequalities.
Where these lines intersect determines the shape of the feasible region, the area where all the inequalities in the system hold true. The corner points are informative because in many optimization problems, such as those in linear programming, the optimal solution lies at one of these vertices. To find these points methodically, one has to calculate the points of intersection between all possible pairs of lines making up the inequalities.
Inequality Graphing
Inequality graphing is a visual way to represent solutions to inequalities or systems of inequalities. It is a crucial step in identifying feasible regions for linear programming problems. When we graph inequalities like those in our exercise, we start by drawing the boundary lines of each inequality. For \(3x - 2y \leq 6\), for instance, we draw the line \(3x - 2y = 6\) and then shade the region that satisfies the inequality
To determine which side of the boundary line to shade, we can pick a test point that's not on the line—often the origin \((0,0)\) if it's not part of the solution set. From there, shading is applied to the half-plane that contains the solutions to the inequality. For our system of inequalities, each inequality contributes its own shaded region and our final solution set is where all individual shaded areas overlap.
To determine which side of the boundary line to shade, we can pick a test point that's not on the line—often the origin \((0,0)\) if it's not part of the solution set. From there, shading is applied to the half-plane that contains the solutions to the inequality. For our system of inequalities, each inequality contributes its own shaded region and our final solution set is where all individual shaded areas overlap.
Linear Programming
Linear programming is a method for finding the maximum or minimum value of a linear function, subject to a set of constraints expressed as inequalities. This technique is widely used in various fields such as economics, business, engineering, and military applications to solve optimization problems.
In the context of our exercise, evaluating \(2x + 5y\) at each corner point helps us find which point provides the optimal solution if, for instance, \(2x + 5y\) represents a profit function we're trying to maximize. After plotting the inequalities and identifying the feasible region, we plug the x and y coordinates of the corner points into the objective function to calculate which vertex provides the best outcome under the given circumstances.
This is what makes the corner points so significant in linear programming; they are the potential candidates for the optimal solution.
In the context of our exercise, evaluating \(2x + 5y\) at each corner point helps us find which point provides the optimal solution if, for instance, \(2x + 5y\) represents a profit function we're trying to maximize. After plotting the inequalities and identifying the feasible region, we plug the x and y coordinates of the corner points into the objective function to calculate which vertex provides the best outcome under the given circumstances.
This is what makes the corner points so significant in linear programming; they are the potential candidates for the optimal solution.
Other exercises in this chapter
Problem 115
Sketch the graph of the solution set for the following system of inequalities: $$\left\\{\begin{array}{l}y \geq n x+b(n0) \\\y \leq m x+b(m>0, b>0)\end{array}\r
View solution Problem 116
Will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{aligned}x+y & \geq 6 \\\x & \leq 8
View solution Problem 118
Will help you prepare for the material covered in the next section. Bottled water and medical supplies are to be shipped to survivors of an earthquake by plane.
View solution Problem 114
Write a system of inequalities whose solution set includes every point in the rectangular coordinate system.
View solution