Algebra Exam 6

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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

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)}

Use Gauss-Jordan elimination to solve the system.

-x - y - z = 1 4x + 5y = 0 y - 3z = 0

A. {(14, -10, -3)}

B. {(10, -2, -6)}

C. {(15, -12, -4)}

D. {(11, -13, -4)}

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)}

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)}

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

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)}

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

8x + 5y + 11z = 30 -x - 4y + 2z = 3 2x - y + 5z = 12

A. {(3 - 3t, 2 + t, t)}

B. {(6 - 3t, 2 + t, t)}

C. {(5 - 2t, -2 + t, t)}

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

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)}

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)}

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)}

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)}

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)}

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

A. {(1, 3, 2, 1)}

B. {(1, 4, 3, -1)}

C. {(1, 5, 1, 1)}

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

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

2x + 3y - 5z = 15 x + 2y - z = 4

A. {(6t + 28, -7t - 6, t)}

B. {(7t + 18, -3t - 7, t)}

C. {(7t + 19, -1t - 9, t)}

D. {(4t + 29, -3t - 2, t)}

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

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

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

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

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

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

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)}

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)}

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

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

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)}

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)}

Locate the foci and find the equations of the asymptotes. 4y2 – x2 = 1

A. (0, ±√4/2); asymptotes: y = ±1/3x

B. (0, ±√5/2); asymptotes: y = ±1/2x

C. (0, ±√5/4); asymptotes: y = ±1/3x

D. (0, ±√5/3); asymptotes: y = ±1/2x

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)

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)

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

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

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)}

Find the focus and directrix of each parabola with the given equation. x2 = -4y

A. Focus: (0, -1), directrix: y = 1

B. Focus: (0, -2), directrix: y = 1

C. Focus: (0, -4), directrix: y = 1

D. Focus: (0, -1), directrix: y = 2

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

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

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)

A. x2/43 + y2/28 = 1

B. x2/33 + y2/49 = 1

C. x2/53 + y2/21 = 1

D. x2/13 + y2/39 = 1

Locate the foci and find the equations of the asymptotes. x2/100 - y2/64 = 1

A. Foci: ({= ±2√21, 0); asymptotes: y = ±2/5x

B. Foci: ({= ±2√31, 0); asymptotes: y = ±4/7x

C. Foci: ({= ±2√41, 0); asymptotes: y = ±4/7x

D. Foci: ({= ±2√41, 0); asymptotes: y = ±4/5x

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

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

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)

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

Find the focus and directrix of the parabola with the given equation. 8x2 + 4y = 0

A. Focus: (0, -1/4); directrix: y = 1/4

B. Focus: (0, -1/6); directrix: y = 1/6

C. Focus: (0, -1/8); directrix: y = 1/8

D. Focus: (0, -1/2); directrix: y = 1/2

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)

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

B. (x - 7)2/5 + (y - 6)2/6 = 1

C. (x - 7)2/4 + (y - 6)2/9 = 1

D. (x - 5)2/4 + (y - 4)2/9 = 1

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)

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

Convert each equation to standard form by completing the square on x and y. 9x2 + 25y2 - 36x + 50y - 164 = 0

A. (x - 2)2/25 + (y + 1)2/9 = 1

B. (x - 2)2/24 + (y + 1)2/36 = 1

C. (x - 2)2/35 + (y + 1)2/25 = 1

D. (x - 2)2/22 + (y + 1)2/50 = 1

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

Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis vertical with length = 10 Length of minor axis = 4 Center: (-2, 3)

A. (x + 2)2/4 + (y - 3)2/25 = 1

B. (x + 4)2/4 + (y - 2)2/25 = 1

C. (x + 3)2/4 + (y - 2)2/25 = 1

D. (x + 5)2/4 + (y - 2)2/25 = 1

Find the vertices and locate the foci of each hyperbola with the given equation. x2/4 - y2/1 =1

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

Vertices at (3, 0) and (-3 0); foci at (12, 0) and (-12, 0)

C. Vertices at (4, 0) and (-4, 0); foci at (16, 0) and (-16, 0)

D. Vertices at (5, 0) and (-5, 0); foci at (11, 0) and (-11, 0)