9.4 The Intersection of Three Planes
Chapter
Chapter 9
Section
9.4
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?

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.

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?

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.

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.
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.

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?

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?

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

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.

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

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.
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.
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

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

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

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

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

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

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

1.13mins
Q9c

Determine the solution to each system.

1. x-y+z=2

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

3. x+y-z=-2

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

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

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.
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.
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

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

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

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

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

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

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

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

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

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

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.

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.

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

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}}