A system of two equations in three unknowns has been manipulated, and, after correctly using elementary operations, a student arrives at the following equivalent system of equations:

• x - y + z = 1
• 0x + 0y + 0z = 3

a. Explain what this equivalent system means.

b. Give an example of a system of equations that might lead to this solution.

A system of two equations in three unknowns has been manipulated, and, after correctly using elementary operations, a student arrives at the following equivalent system of equations:

1. 2x - y + 2z = 1

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

a. Write a solution to this system of equations, and explain what your solution means.

b. Give an example of a system of equations that leads to your solution in part a.

A system of two equations in three unknowns has been manipulated, and, after correctly using elementary operations, a student arrives at the following equivalent system of equations:

1. x - y + z = -1

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

a. Write a solution to this system of equations, and explain what your solution means.

b. Give an example of a system of equations that leads to your solution in part a.

Consider the following system of equations:

1. 2x + y + 6z = p

2. x + my + 3z = q

a) Determine values of m, p, and q such that the two planes are coincident. Are these values unique? Explain.

Consider the following system of equations:

1. 2x + y + 6z = p

2. x + my + 3z = q

b) Determine values of m, p, and q such that the two planes are parallel and not coincident. Are these values unique? Explain.

Consider the following system of equations:

1. 2x + y + 6z = p

2. x + my + 3z = q

• A value of m such that the two planes intersect at right angles. Is this value unique? Explain.

• Determine values of m, p, and q such that the two planes intersect at right angles. Are these values unique? Explain.

Consider the following system of equations:

1. x + 2y - 3z = 0

2. y + 3z = 0

a. Solve this system of equations by letting z = s.

Consider the following system of equations:

1. x + 2y - 3z = 0

2. y + 3z = 0

b. Solve this system of equations by letting y = t.

Consider the following system of equations:

1. x + 2y - 3z = 0

2. y + 3z = 0

Show that the solution you found in part a. is the same as the solution you found in part b.

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. x + y + z = 1

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

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. 2x - y +z +1 = 0

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

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. x - y + 2z = 2

2. x + y + 2z = -2

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. x + y + 2z = 4

2. x - y = 6

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. 2x - y + 2 = 2

2. -x + 2y + z = 1

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. x - y + 2z = 0

2. z = 4

A system of equations is given as follows:

1. x + y + 2z = 1

2. kx + 2y + 4z = k

a. For what value of k does the system have an infinite number of solutions? Determine the solution to the system for this value of k.

b. Is there any value of k for which the system does not have a solution? Explain.

Determine the vector equation of the line that passes through A(-2,3,6) and is parallel to the line of intersection of the planes \pi_1: 2x - y + z = 0 and \pi_2: y + 4z =0.

For the panes 2x - y + 2z = 0 and 2x + y + 6z = 4, show that their line of intersection lies on the plane with equation 5x + 3y + 16z - 11 = 0.

The line of intersection of the planes \pi_1: 2x + y - 3z = 3 and \pi_2: x - 2y + z = -1 is L.

a) Determine parametric equations for L.

The line of intersection of the planes \pi_1: 2x + y - 3z = 3 and \pi_2: x - 2y + z = -1 is L.

b) If L meets the xy-plane at point A and the z-axis at point B, determine the length of line segment AB.

Determine the Cartesian equation of the plane that is parallel to the line with equation x = - 2y = 3z and that contains the line of intersection of the planes with equations x - y + z = 1 and 2y - z = 0.