ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing, graphing, and substitution. (C) Graphing, substitution, and addition. (D) Addition and graphing. (E) Substitution and addition. Replacing is not a method for solving systems of equations. Solving and replacing are not methods for solving systems of equations. C. Correct! This answer choice lists the methods for solving a system of equations. This list is missing a method for solving systems of equations. This list is missing a method for solving systems of equations. The methods to solve a system of equations are graphing, substitution, and addition. These solving methods were discussed in the tutorial with the aid of examples. (C) Graphing, substitution, and addition.
No. 2 of 10 2. A system of equations is dependent if. (A) The equations share no solutions. (B) The equations share all of their solutions. (C) The equations share exactly two solutions. (D) The equations share some of the same solutions. (E) The equations share exactly one solution. Review the definition of dependent systems of equations and try again. B. Correct! This is the correct definition of a dependent system. Review the definition of dependent systems of equations and try again. Review the definition of dependent systems of equations and try again. Review the definition of dependent systems of equations and try again. 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. (B) The equations share all of their solutions.
No. 3 of 10 3. Find the solution to the system of equations formed by y = -2x and y = -3x + 1. (A) (-1, 2) (B) (2, -5) (C) (0, 1) (D) (1, -2) (E) (0, 0) This solution only returns a true statement in the first equation. This solution only returns a true statement in the second equation. This solution only returns a true statement in the second equation. D. Correct! This solution returns a true statement for both of the equations. This solution only returns a true statement in the first equation. The solution (1, -2) can be found using substitution. Substitute y = -2x for y in the second equation. -2x = -3x + 1 x = 1 Substitute x = 1 into y = -2x to find the value of y. y = -2(1) = -2 (D) (1, -2)
No. 4 of 10 4. What is the solution of a system of equations using the graphing method? (A) The origin. (B) The highest point on the graph. (C) The intersection point of the equations. (D) The point where each graph crosses the x-axis. (E) The point where each graph crosses the y-axis. The origin is not always the solution of a system of equations. The point of intersection is where the solution to a system of equations is found. C. Correct! The solution of a system of equations is a point or points where all of the equations in the system are true. The solution of a system of equations is a point or points where all of the equations in the system are true. The solution of a system of equations is a point or points where all of the equations in the system are true. The intersection point of the equations in a system is the solution to that system. (C)The intersection point of the equations.
No. 5 of 10 5. What is the system solving method that uses elimination to solve a system of equations? (A) Addition (B) Graphing (C) Substitution (D) Replacing (E) Subtraction A. Correct! The addition method uses elimination of variables to solve a system of equations. The graphing method uses graphing on the coordinate plane to solve. Substitution involving replacing equations to solve for the other variables. This does not name one of the solving methods discussed in the tutorial. This does not name one of the solving methods discussed in the tutorial. Addition is the system of equations 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 so one variable has opposite coefficients to eliminate one variable. (A) Addition
No. 6 of 10 6. Solve the system of equations formed by 3x + y = 5 and 6x y = 4. (A) (0, 5) (B) (1, 2) (C) (-3, 4) (D) (2, 8) (E) (2, -1) B. Correct! This solution returns a true statement in both equations. This solution returns a false statement in both equations. This solution returns a false statement in the first equation. Point (1,2) is a solution to this system of equations and can be found using the addition method. Add 3x + y = 5 and 6x y = 4 to get 9x = 9. Use x = 1 to find the value of y in 3x + y = 5 to get 3(1) + y = 5. Solve for y to get y = 2. Point (1, 2) is the solution to this system of equations. (B) (1, 2)
No. 7 of 10 7. Solve the system of equations consisting of 3x 2y = 7 and 5x + 4y = 8 using the elimination method. (A) (3, 0.5) (B) (2, -0.5) (C) (6, -2) (D) (0, 2) (E) (1, -2) B. Correct! This solution returns a true statement in both equations. This solution returns a false statement in both equations. This solution returns a false statement in the first equation. 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 = -0.5. The solution is (2, -0.5). (B) (2, -0.5)
No. 8 of 10 8. A system of linear equations is independent if: (A) The equations share exactly one solution. (B) The equations share all of their solutions. (C) The equations share exactly two solutions. (D) The equations share some of the same solutions. (E) The equations share no solutions. A. Correct! A system of linear equations is independent when the equations share exactly one solution. Review the definition of an independent system. Review the definition of an independent system. Review the definition of an independent system. Review the definition of an independent system. A system of linear equations is independent when the equations share exactly one solution. (A) The equations share exactly one solution.
No. 9 of 10 9. Solve the system of equations formed by y = 2x + 3 and 5y 7x = 18 by substitution. (A) (6, 12) (B) (4, 11) (C) (1, 5) (D) (2, 7) (E) (11, 19) This solution returns a false statement in the first equation. C. Correct! This solution returns a true statement in both equations. This solution returns a false statement in the first equation. Substitute y = 2x + 3 into 5y 7x = 18. Simplify 5(2x + 3) 7x = 18 to get 10x + 15-7x = 18. Solve 3x + 15 = 18 to get x = 1. Substitute x = 1 into y = 2x + 3 to get y = 5. Hence, (1, 5) is the solution to this system of equations. (C) (1, 5)
No. 10 of 10 10. What is the solution to a system of parallel lines? (A) One solution (B) Two solutions (C) Infinitely many solutions (D) No solution (E) Cannot be determined Remember that parallel lines never intersect. B. Incorrect Remember that parallel lines never intersect. Remember that parallel lines never intersect. D. Correct! Since parallel lines never intersect, this system has no solutions. Remember that parallel lines never intersect. The solution to a system of linear inequalities is the point where all of the equations in the system intersect. Parallel lines never intersect so a system of parallel lines has no solution. (D) No solution