Questions and Answers
Which of the following best describes a system of two equations where the lines intersect?
Which of the following best describes a system of two equations where the lines are parallel?
How many solutions does a system of two equations have if the lines are coincident?
infinite
Which of the following best describes the solutions to a system of inequalities that continue infinitely in one direction?
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In a linear programming problem, if an optimal objective value exists, where in the feasible solution region will the optimal objective value be located?
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Which of the following points satisfies the inequality 5x - 3y ≥ -6?
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Solve by elimination: 2x - 6y = 5 and 3x + 6y = 5.
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Solve by elimination: 2x + 3y = 9 and x - 2y = -6.
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Solve by substitution: 2x + y = 10 and 3x - 2y = 1.
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Solve by substitution: x - y = -2 and 3x - 4y = -8.
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Elementary Row Operations: Perform the operations and write the resulting system: x + y - z = 8, 2x - y + 3z = 9, x + 2y + z = 5.
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Elementary Row Operations: Perform the operations and write the resulting system: x + y - z = 8, 2x - y + 3z = 9, x + 2y + z = 5.
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Solve the three-variable system of equations: -3x + 7y + 6z = -20, x - 3y + 2z = -6, -2x + 5y + 5z = -16.
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Graph the two-variable inequality: 4x - y ≤ 4.
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Graph the two-variable inequality: y > x(2) + 3.
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Write the vertex of the intersection region of the system of inequalities of the given graph.
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State if the intersection region of the system is bounded or unbounded.
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Write the objective function for the library's collection.
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List the three constraints for the library's book purchase.
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What is the name of the shaded region in the graph for the farmer's market manager?
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What are the vertices in the graph for the farmer's market manager?
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What is the objective function for the farmer's market manager?
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What is the optimal value of profit for the farmer's market manager?
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How many pints of blueberries and strawberries should the manager purchase in order to optimize his profit?
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Study Notes
Systems of Equations
- Consistent systems have intersecting lines, indicating one unique solution.
- Inconsistent systems feature parallel lines with no solutions.
- Coincident lines indicate an infinite number of solutions in the system.
Inequalities
- An unbounded region refers to a solution set that extends infinitely in at least one direction.
- The solution to a system of inequalities may create a feasible region that can be bounded or unbounded.
Linear Programming
- Optimal objective values are found at the vertices of the feasible solution region.
- Objective function example: For a library adding dictionaries and thesauruses, express cost as ( C = 15x + 10y ).
Constraints in Problems
- Constraints for library book additions include:
- ( x + y \leq 20 ) (total books),
- ( x \leq 16 ) (dictionaries),
- ( y \leq 9 ) (thesauruses).
Graphing Linear Inequalities
- Graphs for inequalities should visually represent the constraints and the feasible region.
- Graphing equations helps identify bounded vs. unbounded regions and optimal solutions.
Three Variable Systems
- Example solution for a system of three equations: ( (3, 1, -3) ).
Vertex and Region Classification
- The intersection region of a system of inequalities can be classified as unbounded or bounded.
- Example vertex of intersection for inequalities might be at ( (2,2) ).
Farmer’s Market Problem
- Maximum capacity for berries is 50 pints; limits for blueberries (25 pints) and strawberries (40 pints).
- Objective function for profit is defined as ( P = 3x + 5y ).
- Optimal profit value calculated at ( $230 ).
- To maximize profit, purchase 10 pints of blueberries and 40 pints of strawberries.
- Vertices for the purchase options include ( (0,0), (10,40), (25,25), (0,40), (25,0) ).
Solving Techniques
- Systems can be solved using elimination or substitution methods.
- Example solutions using elimination yield coordinates such as ( (2, -\frac{1}{6}), (0, 3) ).
- Use graphing methods to visualize solutions and intersections in linear programming scenarios.
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Description
This quiz covers systems of equations, inequalities, and linear programming. It focuses on concepts such as consistent and inconsistent systems, unbounded regions, and constraints in problem-solving. You'll explore how to apply these principles in practical scenarios like library book additions.
