9.4 The Intersection of Three Planes
Chapter
Chapter 9
Section
9.4
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Lectures 4 Videos
Solutions 39 Videos

A student is manipulating a system of equations and obtains the following equivalent system:

1. x - 3y + z =2

2. 0x + y -z =-1

3. 0x + 0y + 3z = -12

a. Determine the solution to this system of equations.

b. How would your solution be interpreted geometrically?

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1.44mins
Q1

When manipulating a system of equations, a student obtains the following equivalent system:

1. x - y + z = 4

2. 0x + 0y + 0z = 0

3. 0x + 0y + 0z = 0

a. Give a system of equations that would produce this equivalent system.

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0.56mins
Q2a

When manipulating a system of equations, a student obtains the following equivalent system:

1. x - y + z = 4

2. 0x + 0y + 0z = 0

3. 0x + 0y + 0z = 0

b. How would you interpret the solution to this system geometrically?

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0.25mins
Q2b

When manipulating a system of equations, a student obtains the following equivalent system:

1. x - y + z = 4

2. 0x + 0y + 0z = 0

3. 0x + 0y + 0z = 0

c. Write the solution to this system using parameters for x and y.

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0.56mins
Q2c

When manipulating a system of equations, a student obtains the following equivalent system:

1. x - y + z = 4

2. 0x + 0y + 0z = 0

3. 0x + 0y + 0z = 0

• Write the solution to this system using parameters for y and z.
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0.55mins
Q2d

When manipulating a system of equations, a student obtains the following equivalent system:

1. 2x - y + 3z = -2

2. x - y +4z=3

3. 0x+0y+0z=1

a) Give two systems of equations that could have produced this result.

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0.57mins
Q3a

When manipulating a system of equations, a student obtains the following equivalent system:

1. 2x - y + 3z = -2

2. x - y +4z=3

3. 0x+0y+0z=1

b) What does this equivalent system tell you about possible solutions for the original system of equations?

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1.17mins
Q3b

When manipulating a system of equations, a student obtains the following equivalent system:

1. x + 2y - z =4

2. x+0y-2z=0

3. 2x+0y+0z=-6

a) Without using any further elementary operations, determine the solution to this system.

b) How can the solution to this system be interpreted geometrically?

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2.04mins
Q4

a) Without solving the following system, how can you deduce that these three planes must intersect in a line?

1. 2x-y+z=1

2. x+y-z=-1

3. -3x-3y+3z=3

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0.49mins
Q5a
1. 2x-y+z=1

2. x+y-z=-1

3. -3x-3y+3z=3

b) Find the solution to the given system using elementary operations.

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1.58mins
Q5b

Explain why there is no solution to the following system of equations:

1. 2x+3y-4z=-5

2. x-y+3z=-201

3. 5x-5y+15z=-1004

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1.23mins
Q6

Avery is solving a system of equations using elementary operations and derives, as one of the equations, 0x+0y+0z=0.

• Is it true that this equation will always have a solution? Explain.
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0.30mins
Q7a

Avery is solving a system of equations using elementary operations and derives, as one of the equations, 0x+0y+0z=0.

• Construct your own system of equations in which the equation 0x+0y+0z=0 appears, but for which there is no solution to the constructed system of equations.
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1.00mins
Q7b

Solve the following systems of equations using elementary operations. Interpret your results geometrically.

1. 2x+y-z=-3

2. x-y+2z=0

3. 3x+2y-z=-5

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3.10mins
Q8a

Solve the following systems of equations using elementary operations. Interpret your results geometrically.

1. \displaystlye{\frac{x}{3}}-\displaystyle{\frac{y}{4}}+z = \displaystyle{\frac{7}{8}}

2. 2x+2y-3z=-20

3. x-2y+3z=2

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4.28mins
Q8b

Solve the following systems of equations using elementary operations. Interpret your results geometrically.

1. x-y=-199

2. x+z=-200

3. y-z=201

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1.57mins
Q8c

Solve the following systems of equations using elementary operations. Interpret your results geometrically.

1. x-y-z=-1

2. y-2=0

3. x+1=5

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1.48mins
Q8d

Solve each system of equations using elementary operations. Interpret your results geometrically.

1. x-2y+z=3

2. 2x+3y-z=-9

3. 5x-3y+2z=0

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4.25mins
Q9a

Solve each system of equations using elementary operations. Interpret your results geometrically.

1. x-2y+z=3

2. x+y+z=2

3. x-3y+z=-6

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2.27mins
Q9b

Solve each system of equations using elementary operations. Interpret your results geometrically.

c)

1. x-y+z=-2

2. x+y+z=2

3. x-3y+z=-6

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1.13mins
Q9c

Determine the solution to each system.

1. x-y+z=2

2. 2x-2y+2z=4

3. x+y-z=-2

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1.44mins
Q10a

Determine the solution to each system.

b)

1. 2x-y+3z=0

2. 4x-2y+6z=0

3. -2x+y-3z=0

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0.57mins
Q10b

a) Use elementary operations to show that the following system does not have a solution:

1. x+y+z=1

2. x-2y+z=0

3. x-y+z=0

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1.31mins
Q11a
1. x+y+z=1

2. x-2y+z=0

3. $x-y+z=0$

• Calculate the direction vectors for the lines of intersection between each pair of planes.
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1.41mins
Q11b
1. x+y+z=1

2. x-2y+z=0

3. $x-y+z=0$

• Explain, in your own words, why the planes represented in this system of equations must correspond to a triangular prism.
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2.06mins
Q11cd

Each of the following systems does not have a solution. Explain why.

1. x-y+3z=3

2. x-y+3z=6

3. 3x-5z=0

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0.28mins
Q12a

Each of the following systems does not have a solution. Explain why.

b)

1. 5x-2y+3z=1

2. 5x-2y+3z=-1

3. 5x-2y+3z=13

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0.21mins
Q12b

Each of the following systems does not have a solution. Explain why.

c)

1. x-y+z=9

2. 2x-2y+2z=19

3. 2x-2y+2z=17

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0.31mins
Q12c

Each of the following systems does not have a solution. Explain why.

1. 3x-2y+z=4

2. 9x-6y+3z=12

3. 6x-4y+2z=5

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0.32mins
Q12d

Determine the solution to each system of equations, if a solution exists.

1. 2x-y-z=10

2. x+y+0z=7

3. 0x+y-z=8

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2.15mins
Q13a

Determine the solution to each system of equations, if a solution exists.

b)

1. 2x-y+z=-3

2. x+y-2z=1

3. 5x+2y-5z=0

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3.12mins
Q13b

Determine the solution to each system of equations, if a solution exists.

c)

1. x+y-z=0

2. 2x-y+z=0

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

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2.09mins
Q13c

Determine the solution to each system of equations, if a solution exists.

d)

1. x-10y+13z=-4

2. 2x-20y+26z=-8

3. x-10y+13z=-8

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0.26mins
Q13d

Determine the solution to each system of equations, if a solution exists.

e)

1. x-y+z=-2

2. x+y+z=2

3. 3x+y+3z=2

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1.26mins
Q13e

Determine the solution to each system of equations, if a solution exists.

f)

1. x+y+z=0

2. x-2y+3z=0

3. 2x-y+3z=0

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1.14mins
Q13f

The following system of equations represents three planes that intersect in a line:

1. 2x+y+z=4

2. x-y+z=p

3. 4x+qy+z=2

a) Determine p and q.

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3.52mins
Q14a

The following system of equations represents three planes that intersect in a line:

1. 2x+y+z=4

2. x-y+z=p

3. 4x+qy+z=2

b) Determine an equation in parametric form for the line of intersection.

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1.11mins
Q14b

Consider the following system of equations:

1. 4x+3y+3z=-8

2. 2x+y+z=-4

3. 3x-2y+(m^2-6)z = m-4

Determine the value(s) of m for which this system of equations will have

a) no solution

b) one solution

c.) an infinite number of solutions

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4.29mins
Q15

Determine the solution to the following system of equations:

1. \displaystyle{\frac{1}{a}}+\displaystyle{\frac{1}{b}}-\displaystyle{\frac{1}{c}}=0

2. \displaystyle{\frac{2}{a}}+\displaystyle{\frac{3}{b}}+ \displaystyle{\frac{2}{c}}=\displaystyle{\frac{13}{6}}

3. \displaystyle{\frac{4}{a}}-\displaystyle{\frac{2}{b}}+\displaystyle{\frac{3}{c}}=\displaystyle{\frac{5}{2}}

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5.40mins
Q16