CLEP Precalculus - Problem Drill 15: Systems of Equations and Inequalities
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1 CLEP Precalculus - Problem Drill 15: Systems of Equations and Inequalities No. 1 of What are the methods to solve a system of equations? (A) Graphing, replacing, substitution and matrix techniques. (B) Solving, replacing, graphing and substitution. (C) Graphing, substitution, addition and matrix techniques. (D) Solving and graphing. (E) Completing the square and replacing. Substitution and replacing characterize the same technique. Think about other techniques presented to solve a system of equations. B. Incorrect! Substitution and replacing characterize the same technique. Think about other techniques presented to solve a system of equations. C. Correct! This choice lists the methods presented to solve a system of equations. This answer choice is missing two of the methods. Recall the other solving methods that were discussed in the tutorial. Neither of these methods was presented in the lesson. Think about the techniques necessary to solve a system of equations. The methods to solve a system of equations are: graphing, substitution, addition, and matrix techniques. These solving methods were discussed in the tutorial with the aid of examples. The correct answer is (C).
2 No. 2 of A system of equations is dependent if: (A) The equations do not share all of the same solutions. (B) All of the solutions to one equation are the solutions to another. (C) The equations share one of the same solutions. (D) The equations share some of the same solutions. (E) The equations are not parent functions. Think about the definition of dependent and independent systems of equations. B. Correct! A system of equations is dependent if all of the solutions to one equation are the solutions to the other equations in the system. These equations will be on the same line. Think about the definition of dependent and independent systems of equations. Think about the definition of dependent and independent systems of equations. Think about the definition of dependent and independent systems of equations. A system of equations is dependent if all of the solutions to one equation are the solutions to the other equations in the system. These equations will be on the same line. The correct answer is (B).
3 No. 3 of Find the solution to the system of equations. y y = 2x = 3x + 1 (A) (1, 2) (B) (3, 6) (C) (5, 0) (D) (1, -2) (E) no solution Use one of the techniques presented in the tutorial to solve this system of equations. B. Incorrect! Use one of the techniques presented in the tutorial to solve this system of equations. Use one of the techniques presented in the tutorial to solve this system of equations. D. Correct! You used one of the techniques presented in the tutorial to solve this system of equations. Use one of the techniques presented in the tutorial to solve this system of equations. (1,-2) can be found by substitution. Solve y = -2x for x. Substitute x = -y/2 for x in y = -3x + 1: y = -3(-y/2) + 1 Simplify: (-1/2)y =1 Solve for y to get: y = -2. Use y = -2 to solve for the variable x using x= -y/2 to get x = 1. The correct answer is (D).
4 No. 4 of What is the solution of a system of equations using the graphing method? (A) The origin. (B) The intersection point of both equations. (C) The point where each graph crosses the x-axis. (D) The point where each graph crosses the y-axis. (E) The area between the graphs of the equations. Think about the points that a system of equations may have in common. B. Correct! The point of intersection is where the solution to a system of equations is found. Think about the points that a system of equations may have in common. Think about the common point of the equations in the system. Think about the points that a system of equations may have in common. The intersection point of equations in a system of equations is the solution to that system. The correct answer is (B).
5 No. 5 of What is the system solving method that uses elimination to solve a system of equations? (A) addition (B) graphing (C) substitution (D) matrix techniques (E) completing the square A. Correct! Addition is the system solving method that uses elimination of variables to solve a system of equations. B. Incorrect! Think about which system solving method involves the use of multiplying a constant to eliminate one variable. Think about which system solving method involves the use of multiplying a constant to eliminate one variable. Think about which system solving method involves the use of multiplying a constant to eliminate one variable. Think about which system solving method involves the use of multiplying a constant to eliminate one variable. Addition is the system of equations solving method that uses elimination of one variable to solve for the others in the system of equations. The elimination of variables involves multiplying one equation by a constant to eliminate one variable. The correct answer is (A).
6 No. 6 of Solve the system of equations. 3x + y = 5 6x y = 4 (A) (4, 5) (B) (1, 2) (C) (-3, 4) (D) (7, 8) (E) no solution This ordered pair does not satisfy the equations in the given system of equations. B. Correct! (1, 2) is the correct answer because it satisfies the two equations in the given system of equations. This ordered pair does not satisfy the equations in the given system of equations. This ordered pair does not satisfy the equations in the given system of equations. A solution exists for this system of equations. Use one of the solving techniques to solve these systems of equations. The solution to this system of equations can be found using the addition method. Add 3x + y = 5 and 6x y = 4 to get 9x = 9 Next, use x = 1 to find the value of y in 3x + y = 5 to get 3(1) + y = 5. Solve 3(1) + y = 5 for y to get y = 2. Therefore, (1, 2) is the solution to this system of equations. The correct answer is (B).
7 No. 7 of Solve the system of equations using the addition method. 3x 2y = 7 5x + 4y = 8 (A) ( 1 3, 2 ) (B) ( 1 2, 2 ) (C) (6, -2) (D) (-7, 5) (E) ( 1 4, 5 ) Review the steps for solving a system using the addition method and try again. B. Correct! You multiplied one of the equations by a constant and eliminated one of the variables to solve. Review the steps for solving a system using the addition method and try again. Review the steps for solving a system using the addition method and try again. Review the steps for solving a system using the addition method and try again. (2, -1/2) is the correct answer. Multiply 3x 2y = 7 by 2 to get 6x 4y = 14. Add 6x 4y = 14 to 5x + 4y = 8 to get 11x = 22. Solve 11x = 22 for x to get x = 2. Plug x = 2 into equation 3x 2y = 7 to get 3(2) 2y = 7. Simplify and solve for y: 6 2y = 7, y = -1/2. The correct answer is (B).
8 No. 8 of A system of equations is independent if: (A) There is exactly one solution. (B) There are an infinite number of solutions. (C) The graphs of the equations are parallel lines. (D) The equations graph the same line. (E) There is no solution. A. Correct! A system of equations is independent because the equations do not share all of the same solutions. B. Incorrect! Review the definition of an independent system of equations and try again. Review the definition of an independent system of equations and try again. Review the definition of an independent system of equations and try again. Review the definition of an independent system of equations and try again. A system of equations is independent when it has exactly one solution. The correct answer is (A).
9 No. 9 of Solve the system of equations using the substitution method. y = 2x + 3 7x + 5y = 18 (A) (3, -2) (B) (4, 11) (C) (1, 5) (D) (-8, 7) (E) no solution Review the steps for solving a system using the substitution method and try again. B. Incorrect! Review the steps for solving a system using the substitution method and try again. C. Correct! This point satisfies both equations in the given system of equations. Review the steps for solving a system using the substitution method and try again. Review the steps for solving a system using the substitution method and try again. (1,5) is the correct answer. Replace y in -7x + 5y = 18 with y = 2x + 3. Simplify: -7x + 5(2x + 3) = 18 to get -7x + 10x + 15 = 18. Solve 3x + 15 = 18 to get x = 1. Use x = 1 in y = 2x + 3 to get y = 5. Hence, (1, 5) is the solution to this system of equations. The correct answer is (C).
10 No. 10 of What is the solution to a system of inequalities? (A) The intersection of the two lines. (B) The endpoints of each inequality. (C) All points in the coordinate plane. (D) Their shared shaded region. (E) All points in quadrant I. Recall the definition of a system of inequalities and try again. B. Incorrect Recall the definition of a system of inequalities and try again. Recall the definition of a system of inequalities and try again. D. Correct! The solution to a system of linear inequalities is the area where both inequality graphs overlap. Recall the definition of a system of inequalities and try again. The solution to a system of linear inequalities is the area where both inequality graphs overlap. The correct answer is (D).
A. Incorrect! Replacing is not a method for solving systems of equations.
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