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Chapter 8: Problem 10
Graph the solution set to each system of inequalities and identify each vertexof the region. $$\begin{aligned}&x \geq 0, y \geq 0\\\&50 x+40 y \leq 200\\\&10 x+20 y \leq60\end{aligned}$$
Short Answer
Expert verified
Vertices are \((0,0)\), \((0,3)\), \((2.4,1.8)\), and \((4,0)\).
Step by step solution
01
Identify the inequalities
The system of inequalities given is: 1. \(x \geq 0\)2. \(y \geq 0\)3. \(50x + 40y \leq 200\)4. \(10x + 20y \leq 60\)
02
Graph each inequality
Start by graphing the boundary lines for each inequality on the Cartesian plane. 1. For \(x \geq 0\), draw a vertical line along the y-axis at \(x = 0\) and shade to the right. 2. For \(y \geq 0\), draw a horizontal line along the x-axis at \(y = 0\) and shade upwards. 3. For \(50x + 40y = 200\), simplify to \(x + 0.8y = 4\). Find intercepts: when \(x = 0\), \(y = 5\); when \(y = 0\), \(x = 4\). Graph this line and shade below it. 4. For \(10x + 20y = 60\), simplify to \(x + 2y = 6\). Find intercepts: when \(x = 0\), \(y = 3\); when \(y = 0\), \(x = 6\). Graph this line and shade below it.
03
Identify the feasible region
The feasible region is where all shaded areas overlap. This region is bounded by the lines and includes intersections of the lines.
04
Find the vertices of the region
Determine the points of intersection of the boundary lines and check where they fall within the feasible region: 1. Intersection of \(x + 0.8y = 4\) and \(x + 2y = 6\): Solve the system of linear equations. Multiply the second equation by 0.8 to make coefficients of y equal:\(x + 0.8y = 4\) and \(0.8x + 1.6y = 4.8\). Solving this system yields \(x = 2.4\) and \(y = 1.8\).2. Intersection with axes: \(x = 0, y = 0\); \(x = 4, y = 0\); \(x = 0, y = 3\). Calculate and check each point.
05
List the vertices
The vertices of the region are at the points: \((0,0)\), \((0,3)\), \((2.4,1.8)\), and \((4,0)\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
graphing inequalities
Graphing inequalities on a Cartesian plane helps identify the region where the solutions to those inequalities lie. To graph an inequality, you start by drawing its boundary line. For a linear inequality like the ones given, convert the inequality into an equality (e.g., turn 50x + 40y ≤ 200 into 50x + 40y = 200). Then find the x- and y-intercepts and draw the line connecting them. The solution to the inequality will be on one side of this line. Use the inequality sign to determine on which side to shade. If it's 'less than or equal to,' like 50x + 40y ≤ 200, shade below the line. If it's 'greater than or equal to,' shade above.
feasible region
The feasible region is the overlapping area that satisfies all inequalities in the system. After graphing each inequality, the feasible region lies where all shaded areas intersect. This region is crucial because it contains all the possible solutions to the system of inequalities. For example, in the given exercise, after graphing the inequalities, we see that the overlapping region is a bounded polygon. Finding this region involves careful graphing and shading of each inequality separately.
vertices of region
Vertices are the corner points of the feasible region. They are crucial as they help in identifying the boundary limits of the feasible area. To find them, you look at where the boundary lines intersect. In the given problem, the vertices were found by determining the points of intersection between the lines. For instance, solving the linear equations x + 0.8y = 4 and x + 2y = 6 yields a vertex at (2.4, 1.8). Other vertices were found at the intersections with the x- and y-axes, giving points like (0,0), (0,3), and (4,0). Each vertex needs to be checked if it lies within the feasible region.
linear equations intersection
The intersection of linear equations is where two lines on a graph meet, representing a point that satisfies both equations. In systems of inequalities, finding these intersections is key to identifying vertices of the feasible region. For the given exercise, x + 0.8y = 4 and x + 2y = 6 intersect at (2.4, 1.8). To find this, we solve the system of equations simultaneously. This can be done by making the coefficients of y or x equal and then subtracting one equation from the other. This process unveils the exact point where the lines cross, defining a boundary of the feasible region.
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