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2. For which values of k the following system of linear equations has

(a) no solutions, (b) exactly one solution, and (c) an infinite number of solutions

kx+y+z=1

x+ky+z=1

x+y+kz=1

I don't understand how to solve for k, when it's in each equation

Questioner:Vanessa

Country:Quebec, Canada

Category:Advanced Math

Private:No

Subject:Linear algebra

Question:

2. For which values of k the following system of linear equations has

(a) no solutions, (b) exactly one solution, and (c) an infinite number of solutions

kx+y+z=1

x+ky+z=1

x+y+kz=1

I don't understand how to solve for k, when it's in each equation

..........................................

Suggestion: You have studied Cramer's rule?

D = the determinant:

| k 1 1 |

| 1 k 1 |

| 1 1 k |

If this is nonzero, we have a unique solution.

= k^3 + 1 + 1 - k - k - k

= k^3 - 3k + 2

= (k - 1)(k^2 + k - 2)

= (k - 1)(k + 2)(k - 1)

So this is zero for k = 1 and k = -2.

If k = 1, the equations are:

x+y+z=1

x+y+z=1

x+y+z=1

What do you think about these?

If k = -2, the equations are:

-2x + y + z =1

x -2y + z =1

x +y -2z =1

Try adding these up. Do you like the result?

What do you think about these?

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If you have NOT studied C's R, then take:

kx+y+z=1 A

x+ky+z=1 B

x+y+kz=1 C

1.1: Eliminate x from A,B: Do A - kB, meaning: multiply B by -k and add that to A. Call this D.

1.2: Eliminate x from A,C: Do A - kC. Call this E

Now you have two equations in y,z, called D,E.

2.1. Eliminate y from D,E. (I leave that to you.)

2.2 You now have an equation in z, that looks like:

(k^3 - 3k + 2) z = (something with k's), so:

(something with k's)

z = --------------------

(k^3 - 3k + 2)

You can figure the rest.

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I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

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