University:
College Entrance Preparation
Course:
ACT Preparation (Mathematics)
Academic year:

2019
Views:

400

Pages:

2
Author:

gaarmaakbn9h

Name: Period: 2.1 WS Systems of Linear Equations: Substitution and Elimination For #1-2, decide whether the values of the variables listed are solutions of the system of equations. 1. 3x + 4y = 4 1/2x - 3y = -1/2 x = 2, y = 1/2 3(2) + 4(1/2) = 4 6 + 2 = 4 8 ≠ 4 No 2. 3x + 3y + 2z = 4 x - 3y + z = 10 5x - 2y - 3z = 8 x = 2, y = -2, z = 2 3(2) + 3(-2) + 2(2) = 4 6 - 6 + 4 = 4 ✓ (2) - 3(-2) + 2 = 10 2 + 6 + 2 = 10 ✓ 5(2) - 2(-2) - 3(2) = 8 10 + 4 - 6 = 8 ✓ Yes Solve each system of equations. If the system has no solution, say that it is inconsistent. 3. 3(3x - 6y = 2) 5x + 4y = 1 9x - 18y = 6 15x + 12y = 3 21x = 12 x = 4/7 (4/7, 5/6) 4. 4(x + 2y = 4) - 2 2x + 4y = 8 -2x - 4y = -8 2x + 4y = 8 0 = 0 Infinite Solutions 5. 2x + 3y = 6 x - y = 1/2 (x - y = 1/2) * -2 2x + 3y = 6 -2x + 2y = -1 5y = 5 y = 1 x - 1 = 1/2 x = 3/2 (3/2, 1) 6. x^2 + y = 16 y + x = 4 x^2 + 4 - x = 16 x^2 - x - 12 = 0 (x - 4)(x + 3) = 0 x = 4, x = -3 y = 4 - 4 = 0 y = 4 - (-3) = 7 (4, 0), (-3, 7) 7. 8x + y^2 = 12 x + y = 3 x = 3 - y 8(3 - y) + y^2 = 12 24 - 8y + y^2 = 12 y^2 - 8y + 12 = 0 (y - 6)(y - 2) = 0 y = 6, y = 2 x = 3 - 6 = -3 x = 3 - 2 = 1 (-3, 6), (1, 2) 8. Solve the system: x^2 + y = 16 y + x = 4 x^2 + 4 - x = 16 x^2 - x - 12 = 0 (x - 4)(x + 3) = 0 x = 4, x = -3 y = 4 - 4 = 0 y = 4 - (-3) = 7 (4, 0), (-3, 7) 9. Solve the system: 8x + y^2 = 12 x + y = 3 x = 3 - y 8(3 - y) + y^2 = 12 24 - 8y + y^2 = 12 y^2 - 8y + 12 = 0 (y - 6)(y - 2) = 0 y = 6, y = 2 x = 3 - 6 = -3 x = 3 - 2 = 1 (-3, 6), (1, 2) 10. Choose True or False for each statement about the parabola with the equation `y^2 - 12x - 2y = 23`. y^2 - 2y + 1 = 12x + 23 + 1 (y - 1)^2 = 12(x + 2) A. The vertex is (1, -2). **False** B. The focus is (1, 1). **True** C. The directrix is x = -5. **True** D. The parabola opens to the left. **True** 11. Determine if each line intersects the parabola shown below in exactly two points. Choose Yes or No. [Image of a parabola and several lines] A. y = 2 **Yes** B. x = 2 **Yes** C. y = -2 **Yes** D. x + y = 2 **Yes** E. x + y = -2 **Yes** 12. The equation for the cross section of a parabolic telescope mirror is `34y = x^2`, measured in centimeters. How far is the focus from the vertex of the cross section? 4p/4 = 34/4 p = 8.5 (x + y - z = 1) * -2 2x + 3y + 2z = 2 -2x - 2y + 2z = -2 2x + 3y + 2z = 2 y + 4z = 0 (5y + 3z = 3) * -1 y + 4z = 0 -5y - 3z = -3 y + 4z = 0 0 ≠ 3 Inconsistent 8. 2x - 2y + 3z = 6 4x - 3y + 2z = 0 (2x - 2y + 3z = 6) * -1 4x - 3y + 2z = 0 -2x + 2y - 3z = -6 4x - 3y + 2z = 0 2y - z = -6 (y - 4z = 7) * 3 y - 4z = 7 3y - 12z = 21 y - 4z = 7 0 ≠ 14 Inconsistent 9. 2x - 2y + 3z = 6 4y - 2z = -3 (2x - 2y + 3z = 6) * -1 4y - 2z = -3 -2x + 2y - 3z = -6 4y - 2z = -3 4y - 2z = -6 (2y - z = -3) * 4 8y - 8z = -12 -8y + 2z = 1 11z = 11 z = 1 2y - 1 = -3 2y = -2 y = -1 2x - 2(-1) + 3(1) = 6 2x + 2 + 3 = 6 2x = 1 x = 1/2 (1/2, -1, 1) 10. The perimeter of a rectangular floor is 90 feet. Find the dimensions of the floor if the length is twice the width. 2x + 2(2x) = 90 2x + 4x = 90 6x = 90 x = 15 2x = 30 Width = 15 Length = 30