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calcvecnelsonCh9.3lecturesHQ
1 Videos

Solutions
21 Videos

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.

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0.57mins

Q1

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:

`2x - y + 2z = 1`

`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.

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1.07mins

Q2

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 +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.

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1.15mins

Q3

Consider the following system of equations:

`2x + y + 6z = p`

`x + my + 3z = q`

a) Determine values of `m, p,`

and `q`

such that the two planes are coincident.
Are these values unique? Explain.

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1.01mins

Q4a

Consider the following system of equations:

`2x + y + 6z = p`

`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.

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1.02mins

Q4b

Consider the following system of equations:

`2x + y + 6z = p`

`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.

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0.43mins

Q4cd

Consider the following system of equations:

`x + 2y - 3z = 0`

`y + 3z = 0`

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

.

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0.48mins

Q5a

Consider the following system of equations:

`x + 2y - 3z = 0`

`y + 3z = 0`

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

.

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1.02mins

Q5b

Consider the following system of equations:

`x + 2y - 3z = 0`

`y + 3z = 0`

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

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0.36mins

Q5c

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.

`x + y + z = 1`

`2x + 2y + 2z = 2`

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0.31mins

Q6a

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.

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

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

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0.23mins

Q6b

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.

`x - y + 2z = 2`

`x + y + 2z = -2`

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1.21mins

Q6c

`x + y + 2z = 4`

`x - y = 6`

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1.07mins

Q6d

`2x - y + 2 = 2`

`-x + 2y + z = 1`

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2.13mins

Q6e

`x - y + 2z = 0`

`z = 4`

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0.34mins

Q6f

A system of equations is given as follows:

`x + y + 2z = 1`

`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.

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2.23mins

Q8

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`

.

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1.26mins

Q9

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`

.

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4.48mins

Q10

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`

.

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2.25mins

Q11a

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.

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2.46mins

Q11b

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`

.

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2.55mins

Q12