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Lectures
4 Videos

This book and the Matrix Method

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1.47mins

This book and the Matrix Method

Introduction of Three Intersecting Planes

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8.44mins

Introduction of Three Intersecting Planes

Three Planes intersecting in Plane

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5.24mins

Three Planes intersecting in Plane

Two Parallel Planes with a Third non Parallel Plane

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2.19mins

Two Parallel Planes with a Third non Parallel Plane

Solutions
39 Videos

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

`x - 3y + z =2`

`0x + y -z =-1`

`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:

`x - y + z = 4`

`0x + 0y + 0z = 0`

`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:

`x - y + z = 4`

`0x + 0y + 0z = 0`

`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:

`x - y + z = 4`

`0x + 0y + 0z = 0`

`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:

`x - y + z = 4`

`0x + 0y + 0z = 0`

`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:

`2x - y + 3z = -2`

`x - y +4z=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:

`2x - y + 3z = -2`

`x - y +4z=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:

`x + 2y - z =4`

`x+0y-2z=0`

`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?

`2x-y+z=1`

`x+y-z=-1`

`-3x-3y+3z=3`

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0.49mins

Q5a

`2x-y+z=1`

`x+y-z=-1`

`-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:

`2x+3y-4z=-5`

`x-y+3z=-201`

`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.

`2x+y-z=-3`

`x-y+2z=0`

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

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3.10mins

Q8a

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

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

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

`x-2y+3z=2`

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4.28mins

Q8b

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

`x-y=-199`

`x+z=-200`

`y-z=201`

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1.57mins

Q8c

`x-y-z=-1`

`y-2=0`

`x+1=5`

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1.48mins

Q8d

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

`x-2y+z=3`

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

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

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4.25mins

Q9a

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

`x-2y+z=3`

`x+y+z=2`

`x-3y+z=-6`

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2.27mins

Q9b

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

c)

`x-y+z=-2`

`x+y+z=2`

`x-3y+z=-6`

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1.13mins

Q9c

Determine the solution to each system.

`x-y+z=2`

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

`x+y-z=-2`

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1.44mins

Q10a

Determine the solution to each system.

b)

`2x-y+3z=0`

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

`-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:

`x+y+z=1`

`x-2y+z=0`

`x-y+z=0`

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1.31mins

Q11a

`x+y+z=1`

`x-2y+z=0`

`$x-y+z=0$

- Calculate the direction vectors for the lines of intersection between each pair of planes.

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1.41mins

Q11b

`x+y+z=1`

`x-2y+z=0`

`$x-y+z=0$

- 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.

`x-y+3z=3`

`x-y+3z=6`

`3x-5z=0`

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0.28mins

Q12a

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

b)

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

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

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

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0.21mins

Q12b

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

c)

`x-y+z=9`

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

`2x-2y+2z=17`

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0.31mins

Q12c

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

`3x-2y+z=4`

`9x-6y+3z=12`

`6x-4y+2z=5`

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0.32mins

Q12d

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

`2x-y-z=10`

`x+y+0z=7`

`0x+y-z=8`

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2.15mins

Q13a

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

b)

`2x-y+z=-3`

`x+y-2z=1`

`5x+2y-5z=0`

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3.12mins

Q13b

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

c)

`x+y-z=0`

`2x-y+z=0`

`4x-5y+5z=0`

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2.09mins

Q13c

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

d)

`x-10y+13z=-4`

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

`x-10y+13z=-8`

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0.26mins

Q13d

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

e)

`x-y+z=-2`

`x+y+z=2`

`3x+y+3z=2`

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1.26mins

Q13e

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

f)

`x+y+z=0`

`x-2y+3z=0`

`2x-y+3z=0`

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1.14mins

Q13f

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

`2x+y+z=4`

`x-y+z=p`

`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:

`2x+y+z=4`

`x-y+z=p`

`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:

`4x+3y+3z=-8`

`2x+y+z=-4`

`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:

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

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

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

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5.40mins

Q16