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Homework answers / question archive / Question 1) Solve the following system of equations using matrices

Question 1) Solve the following system of equations using matrices

Math

Question 1)

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

A. {(3, 1, 0)}

B. {(2, 1, 1)}

C. {(4, 2, 1)}  

D. {(2, 1, 0)}

Question 2

Use Cramer’s Rule to solve the following system.

 

x + 2y + 2z = 5
2x + 4y + 7z = 19
-2x - 5y - 2z = 8

 

A. {(33, -11, 4)}

B. {(13, 12, -3)}

C. {(23, -12, 3)}

D. {(13, -14, 3)}

Question 3

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.

Question 4

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)}

Question 5

Use Cramer’s Rule to solve the following system.
 

 

12x + 3y = 15
2x - 3y = 13

 

A. {(2, -3)}

B. {(1, 3)}

C. {(3, -5)}

D. {(1, -7)}

Question 6

Use Cramer’s Rule to solve the following system.

 

4x - 5y - 6z = -1
x - 2y - 5z = -12
2x - y = 7

 

A. {(2, -3, 4)}

B. {(5, -7, 4)}

C. {(3, -3, 3)}

D. {(1, -3, 5)}

Question 7

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

Question 8

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)}

Question 9

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)}

Question 10

Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.

 

x + 3y = 0
x + y + z = 1
3x - y - z = 11

 

A. {(3, -1, -1)}

B. {(2, -3, -1)}

C. {(2, -2, -4)}

D. {(2, 0, -1)}

Question 11

Use Cramer’s Rule to solve the following system.

 

3x - 4y = 4
2x + 2y = 12

 

A. {(3, 1)}

B. {(4, 2)}

C. {(5, 1)}

D. {(2, 1)}

Question 12

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)}

Question 13

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)}

Question 14

Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.

 

3x + 4y + 2z = 3
4x - 2y - 8z = -4
x + y - z = 3

 

A. {(-2, 1, 2)}

B. {(-3, 4, -2)}

C. {(5, -4, -2)}

D. {(-2, 0, -1)}

Question 15

Use Cramer’s Rule to solve the following system.
 

 

x + 2y = 3
3x - 4y = 4

 

A. {(3, 1/5)}

B. {(5, 1/3)}

C. {(1, 1/2)}

D. {(2, 1/2)}

Question 16

Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.

https://study.ashworthcollege.edu/access/content/group/ddef570e-fc7d-4cb1-bd6e-3d3f5ab3f21b/v9/Images/Exam%20Images/Lesson%202/Lesson%202%20Exam%20Question%2010.JPG

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)}

Question 17

Use Gaussian elimination to find the complete solution to each system.

 

x - 3y + z = 1
-2x + y + 3z = -7
x - 4y + 2z = 0

 

A. {(2t + 4, t + 1, t)}

B. {(2t + 5, t + 2, t)}

C. {(1t + 3, t + 2, t)}

D. {(3t + 3, t + 1, t)}

Question 18

Solve the system using the inverse that is given for the coefficient matrix.

https://study.ashworthcollege.edu/access/content/group/ddef570e-fc7d-4cb1-bd6e-3d3f5ab3f21b/v9/Images/Exam%20Images/Lesson%202/Lesson%202%20Exam%20Question%2010.JPG

2x + 6y + 6z = 8
2x + 7y + 6z =10
2x + 7y + 7z = 9


The inverse of:

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2

2

2

  6

7

7

  6

6

7

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is

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7/2

-1

0

  0

1

-1

  -3

0

1

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A. {(1, 2, -1)}

B. {(2, 1, -1)}

C. {(1, 2, 0)}

D. {(1, 3, -1)}

Question 19

Give the order of the following matrix; if A = [aij], identify a32 and a23.

https://study.ashworthcollege.edu/access/content/group/ddef570e-fc7d-4cb1-bd6e-3d3f5ab3f21b/v9/Images/Exam%20Images/Lesson%206/Left%20Bracket.JPG

1
 
0
 
-2

-5
 
7

  1/2


 
-6

  11

e
 
-∏

  -1/5

https://study.ashworthcollege.edu/access/content/group/ddef570e-fc7d-4cb1-bd6e-3d3f5ab3f21b/v9/Images/Exam%20Images/Lesson%206/Right%20Bracket.JPG

 

 

A. 3 * 4; a32 = 1/45; a23 = 6

 

B. 3 * 4; a32 = 1/2; a23 = -6

 

C. 3 * 2; a32 = 1/3; a23 = -5

 

D. 2 * 3; a32 = 1/4; a23 = 4

Question 20

 

Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.

https://study.ashworthcollege.edu/access/content/group/ddef570e-fc7d-4cb1-bd6e-3d3f5ab3f21b/v9/Images/Exam%20Images/Lesson%202/Lesson%202%20Exam%20Question%2010.JPG

5x + 8y - 6z = 14
3x + 4y - 2z = 8
x + 2y - 2z = 3

 

 

A. {(-4t + 2, 2t + 1/2, t)}

 

B. {(-3t + 1, 5t + 1/3, t)}

 

C. {(2t + -2, t + 1/2, t)}

 

D. {(-2t + 2, 2t + 1/2, t)}

 
 

Question 21

     

Locate the foci of the ellipse of the following equation.

x2/16 + y2/4 = 1

A. Foci at (-2√3, 0) and (2√3, 0)

 

B. Foci at (5√3, 0) and (2√3, 0)

 

C. Foci at (-2√3, 0) and (5√3, 0)

 

D. Foci at (-7√2, 0) and (5√2, 0)

 

Question 22

 
       

Find the vertices and locate the foci of each hyperbola with the given equation.

y2/4 - x2/1 = 1

 

 

A. Vertices at (0, 5) and (0, -5); foci at (0, 14) and (0, -14)

 

B. Vertices at (0, 6) and (0, -6); foci at (0, 13) and (0, -13)

 

C.

Vertices at (0, 2) and (0, -2); foci at (0, √5) and (0, -√5)

 

D. Vertices at (0, 1) and (0, -1); foci at (0, 12) and (0, -12)

Question 23

 

Locate the foci of the ellipse of the following equation.
 
7x2 = 35 - 5y2

 

A. Foci at (0, -√2) and (0, √2)

 

B. Foci at (0, -√1) and (0, √1)

 

C. Foci at (0, -√7) and (0, √7)

 

D. Foci at (0, -√5) and (0, √5)

Question 24

 

Find the standard form of the equation of each hyperbola satisfying the given conditions.

Foci: (-4, 0), (4, 0)
Vertices: (-3, 0), (3, 0)

 

 

A. x2/4 - y2/6 = 1

 

B. x2/6 - y2/7 = 1

 

C. x2/6 - y2/7 = 1

 

D. x2/9 - y2/7 = 1

 
 
 

Question 25

     

Find the standard form of the equation of the following ellipse satisfying the given conditions.

Foci: (-2, 0), (2, 0)
Y-intercepts: -3 and 3

 

A. x2/23 + y2/6 = 1

 

B. x2/24 + y2/2 = 1

 

C. x2/13 + y2/9 = 1

 

D. x2/28 + y2/19 = 1

Question 26

 

Find the standard form of the equation of each hyperbola satisfying the given conditions.

Center: (4, -2)
Focus: (7, -2)
Vertex: (6, -2)

 

A. (x - 4)2/4 - (y + 2)2/5 = 1

 

B. (x - 4)2/7 - (y + 2)2/6 = 1

 

C. (x - 4)2/2 - (y + 2)2/6 = 1

 

D. (x - 4)2/3 - (y + 2)2/4 = 1

Question 27

 

Convert each equation to standard form by completing the square on x or y. Then ?nd the vertex, focus, and directrix of the parabola.

y2 - 2y + 12x - 35 = 0

B. (y - 1)2 = -12(x - 3); vertex: (3, 1); focus: (0, 1); directrix: x = 6

 

C. (y - 5)2 = -14(x - 3); vertex: (2, 1); focus: (0, 1); directrix: x = 6

 

D. (y - 2)2 = -12(x - 3); vertex: (3, 1); focus: (0, 1); directrix: x = 8

 
 
 

Question 28

 
       

Find the solution set for each system by finding points of intersection.

https://study.ashworthcollege.edu/access/content/group/ddef570e-fc7d-4cb1-bd6e-3d3f5ab3f21b/v9/Images/Exam%20Images/Lesson%202/Lesson%202%20Exam%20Question%2010.JPG

x2 + y2 = 1
x2 + 9y = 9

 

 

A. {(0, -2), (0, 4)}

 

B. {(0, -2), (0, 1)}

 

C. {(0, -3), (0, 1)}

 

D. {(0, -1), (0, 1)}

 

Question 29

 
       

Convert each equation to standard form by completing the square on x or y. Then ?nd the vertex, focus, and directrix of the parabola.

x2 - 2x - 4y + 9 = 0

 

 

A. (x - 4)2 = 4(y - 2); vertex: (1, 4); focus: (1, 3) ; directrix: y = 1

 

B. (x - 2)2 = 4(y - 3); vertex: (1, 2); focus: (1, 3) ; directrix: y = 3

 

C. (x - 1)2 = 4(y - 2); vertex: (1, 2); focus: (1, 3) ; directrix: y = 1

 

D. (x - 1)2 = 2(y - 2); vertex: (1, 3); focus: (1, 2) ; directrix: y = 5

 

Question 30

 
       

Locate the foci and find the equations of the asymptotes.
 
x2/9 - y2/25 = 1

A. Foci: ({±√36, 0) ;asymptotes: y = ±5/3x

 

B. Foci: ({±√38, 0) ;asymptotes: y = ±5/3x

 

C. Foci: ({±√34, 0) ;asymptotes: y = ±5/3x

 

D. Foci: ({±√54, 0) ;asymptotes: y = ±6/3x

 

Question 31

     

Find the focus and directrix of each parabola with the given equation.

y2 = 4x

 

A. Focus: (2, 0); directrix: x = -1

 

B. Focus: (3, 0); directrix: x = -1

 

C. Focus: (5, 0); directrix: x = -1

 

D. Focus: (1, 0); directrix: x = -1

Question 32

 

Convert each equation to standard form by completing the square on x and y.

9x2 + 16y2 - 18x + 64y - 71 = 0

 

A. (x - 1)2/9 + (y + 2)2/18 = 1

 

B. (x - 1)2/18 + (y + 2)2/71 = 1

 

C. (x - 1)2/16 + (y + 2)2/9 = 1

 

D. (x - 1)2/64 + (y + 2)2/9 = 1

Question 33

 

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

 

A. y2/6 - x2/9 = 1

 

B. y2/36 - x2/9 = 1

 

C. y2/37 - x2/27 = 1

 

D. y2/9 - x2/6 = 1

Question 34

 

Find the vertex, focus, and directrix of each parabola with the given equation.

(y + 3)2 = 12(x + 1)

 

A. Vertex: (-1, -3); focus: (1, -3); directrix: x = -3

 

B. Vertex: (-1, -1); focus: (4, -3); directrix: x = -5

 

C. Vertex: (-2, -3); focus: (2, -4); directrix: x = -7

 

D. Vertex: (-1, -3); focus: (2, -3); directrix: x = -4

Question 35

 

Convert each equation to standard form by completing the square on x and y.

4x2 + y2 + 16x - 6y - 39 = 0

 

A. (x + 2)2/4 + (y - 3)2/39 = 1

 

B. (x + 2)2/39 + (y - 4)2/64 = 1

 

C. (x + 2)2/16 + (y - 3)2/64 = 1

 

D. (x + 2)2/6 + (y - 3)2/4 = 1

Question 36

 

Locate the foci of the ellipse of the following equation.

25x2 + 4y2 = 100

 

A. Foci at (1, -√11) and (1, √11)

 

B. Foci at (0, -√25) and (0, √25)

 

C. Foci at (0, -√22) and (0, √22)

 

D. Foci at (0, -√21) and (0, √21)

Question 37

 

Find the vertex, focus, and directrix of each parabola with the given equation.

(x + 1)2 = -8(y + 1)

 

A. Vertex: (-1, -2); focus: (-1, -2); directrix: y = 1

 

B. Vertex: (-1, -1); focus: (-1, -3); directrix: y = 1

 

C. Vertex: (-3, -1); focus: (-2, -3); directrix: y = 1

 

D. Vertex: (-4, -1); focus: (-2, -3); directrix: y = 1

 
 
 
 

Question 38

 
       

Find the vertex, focus, and directrix of each parabola with the given equation.

(y + 1)2 = -8x

 

A. Vertex: (0, -1); focus: (-2, -1); directrix: x = 2

 

B. Vertex: (0, -1); focus: (-3, -1); directrix: x = 3

 

C. Vertex: (0, -1); focus: (2, -1); directrix: x = 1

 

D. Vertex: (0, -3); focus: (-2, -1); directrix: x = 5

Question 39

 

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)

 

A. x2/49 + y2/ 25 = 1

 

B. x2/64 + y2/39 = 1

 

C. x2/56 + y2/29 = 1

 

D. x2/36 + y2/27 = 1

 
 

 

Question 40

 

Find the vertex, focus, and directrix of each parabola with the given equation.

(x - 2)2 = 8(y - 1)

 

A. Vertex: (3, 1); focus: (1, 3); directrix: y = -1

 

B. Vertex: (2, 1); focus: (2, 3); directrix: y = -1

 

C. Vertex: (1, 1); focus: (2, 4); directrix: y = -1

 

D. Vertex: (2, 3); focus: (4, 3); directrix: y = -1

 
 
 
 

 

 

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