1. Use Gaussian elimination to find the complete solution to each system.
x1 + 4x2 + 3x3 - 6x4 = 5
x1 + 3x2 + x3 - 4x4 = 3
2x1 + 8x2 + 7x3 - 5x4 = 11
2x1 + 5x2 - 6x4 = 4
A. {(-47t + 4, 12t, 7t + 1, t)}
B. {(-37t + 2, 16t, -7t + 1, t)}
C. {(-35t + 3, 16t, -6t + 1, t)}
D. {(-27t + 2, 17t, -7t + 1, t)}
2. Use Cramer’s Rule to solve the following system.
x + y + z = 0
2x - y + z = -1
-x + 3y - z = -8
A. {(-1, -3, 7)}
B. {(-6, -2, 4)}
C. {(-5, -2, 7)}
D. {(-4, -1, 7)}
3. Use Cramer’s Rule to solve the following system.
2x = 3y + 2
5x = 51 - 4y
A. {(8, 2)}
B. {(3, -4)}
C. {(2, 5)}
D. {(7, 4)}
4. Give the order of the following matrix; if A = [aij], identify a32 and a23.
1
0
-2 -5
7
1/2 ∏
-6
11 e
-∏
-1/5
5. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
2x - y - z = 4
x + y - 5z = -4
x - 2y = 4
A. {(2, -1, 1)}
B. {(-2, -3, 0)}
C. {(3, -1, 2)}
D. {(3, -1, 0)}
6. Use Cramer’s Rule to solve the following system.
x + 2y + 2z = 5
2x + 4y + 7z = 19
-2x - 5y - 2z = 8
7. Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
2w + x - y = 3
w - 3x + 2y = -4
3w + x - 3y + z = 1
w + 2x - 4y - z = -2
8. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + y - z = -2
2x - y + z = 5
-x + 2y + 2z = 1
A. {(0, -1, -2)}
B. {(2, 0, 2)}
C. {(1, -1, 2)}
D. {(4, -1, 3)}
9. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + 2y = z - 1
x = 4 + y - z
x + y - 3z = -2
10. Use Cramer’s Rule to solve the following system.
4x - 5y - 6z = -1
x - 2y - 5z = -12
2x - y = 7
11. If AB = -BA, then A and B are said to be anticommutative.
Are A = 0
1 -1
0 and B = 1
0 0
-1 anticommutative?
A. AB = -AB so they are not anticommutative.
B. AB = BA so they are anticommutative.
C. BA = -BA so they are not anticommutative.
D. AB = -BA so they are anticommutative.
12. Use Cramer’s Rule to solve the following system.
4x - 5y = 17
2x + 3y = 3
A. {(3, -1)}
B. {(2, -1)}
C. {(3, -7)}
D. {(2, 0)}
13. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
3x1 + 5x2 - 8x3 + 5x4 = -8
x1 + 2x2 - 3x3 + x4 = -7
2x1 + 3x2 - 7x3 + 3x4 = -11
4x1 + 8x2 - 10x3+ 7x4 = -10
A. {(1, -5, 3, 4)}
B. {(2, -1, 3, 5)}
C. {(1, 2, 3, 3)}
D. {(2, -2, 3, 4)}
14. Solve the system using the inverse that is given for the coefficient matrix.
2x + 6y + 6z = 8
2x + 7y + 6z =10
2x + 7y + 7z = 9
The inverse of:
2
2
2 6
7
7 6
6
7
is
7/2
-1
0 0
1
-1 -3
0
1
A. {(1, 2, -1)}
B. {(2, 1, -1)}
C. {(1, 2, 0)}
D. {(1, 3, -1)}
15. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x - 2y + z = 0
y - 3z = -1
2y + 5z = -2
16. Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + y + z = 4
x - y - z = 0
x - y + z = 2
17. Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
w - 2x - y - 3z = -9
w + x - y = 0
3w + 4x + z = 6
2x - 2y + z = 3
A. {(-1, 2, 1, 1)}
B. {(-2, 2, 0, 1)}
C. {(0, 1, 1, 3)}
D. {(-1, 2, 1, 1)}
18. Use Gauss-Jordan elimination to solve the system.
-x - y - z = 1
4x + 5y = 0
y - 3z = 0
19. Find values for x, y, and z so that the following matrices are equal.
2x
z y + 7
4 = -10
6 13
4
A. x = -7; y = 6; z = 2
B. x = 5; y = -6; z = 2
C. x = -3; y = 4; z = 6
D. x = -5; y = 6; z = 6
20. Use Cramer’s Rule to solve the following system.
x + y = 7
x - y = 3
A. {(7, 2)}
B. {(8, -2)}
C. {(5, 2)}
D. {(9, 3)}
21. Find the standard form of the equation of the following ellipse satisfying the given conditions.
Foci: (-5, 0), (5, 0)
Vertices: (-8, 0), (8, 0)
22. Find the focus and directrix of each parabola with the given equation.
x2 = -4y
23. Find the standard form of the equation of each hyperbola satisfying the given conditions.
Foci: (-4, 0), (4, 0)
Vertices: (-3, 0), (3, 0)
24. Find the vertex, focus, and directrix of each parabola with the given equation.
(y + 1)2 = -8x
25. Find the solution set for each system by finding points of intersection.
x2 + y2 = 1
x2 + 9y = 9
26. Find the vertex, focus, and directrix of each parabola with the given equation.
(y + 3)2 = 12(x + 1)
27. Find the standard form of the equation of each hyperbola satisfying the given conditions.
Center: (4, -2)
Focus: (7, -2)
Vertex: (6, -2)
28. Find the vertices and locate the foci of each hyperbola with the given equation.
y2/4 - x2/1 = 1
29. Find the standard form of the equation of the ellipse satisfying the given conditions.
Endpoints of major axis: (7, 9) and (7, 3)
Endpoints of minor axis: (5, 6) and (9, 6)
30. Find the focus and directrix of the parabola with the given equation.
8x2 + 4y = 0
31. Locate the foci and find the equations of the asymptotes.
x2/9 - y2/25 = 1
32. Find the standard form of the equation of each hyperbola satisfying the given conditions.
Endpoints of transverse axis: (0, -6), (0, 6)
Asymptote: y = 2x
33. Convert each equation to standard form by completing the square on x and y.
34. Find the standard form of the equation of the following ellipse satisfying the given conditions.
Foci: (0, -4), (0, 4)
Vertices: (0, -7), (0, 7)
35. Locate the foci of the ellipse of the following equation.
7x2 = 35 - 5y2
36. Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola.
y2 - 2y + 12x - 35 = 0
37. Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola.
x2 - 2x - 4y + 9 = 0
38. Find the focus and directrix of each parabola with the given equation.
y2 = 4x
39. Locate the foci of the ellipse of the following equation.
x2/16 + y2/4 = 1
40. Find the standard form of the equation of each hyperbola satisfying the given conditions.
Foci: (0, -3), (0, 3)
Vertices: (0, -1), (0, 1)
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