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?

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.

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?

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.

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.

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.

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?

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?

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

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.

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

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.

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.

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

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

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

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

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

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

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

Determine the solution to each system.

1. x-y+z=2

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

3. x+y-z=-2

Determine the solution to each system.

b)

1. 2x-y+3z=0

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

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

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. 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. 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.

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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}}