Correct me if I am wrong. Find the value(s) of the constant k such that...

Makayla Boyd

Makayla Boyd

Answered question

2022-06-21

Correct me if I am wrong. Find the value(s) of the constant k such that the system of linear equations
{ x + 2 y = 1 k 2 x 2 k y = k + 2
has:
1. No solution
2. An infinite number of solutions
3. Exactly one solution
Answer:
so the first step is to get row reduction form, which is:
from [ 1 2 k 2 2 k ] ,
to [ 1 2 0 2 k + 2 k 2 ]

Answer & Explanation

Jayce Bates

Jayce Bates

Beginner2022-06-22Added 18 answers

You row reduction is wrong. We get
1 2 2 0 2 k 2 2 k k 2 + k + 2
which is equivalent to
1 2 2 0 2 k ( k + 1 ) ( k + 1 ) ( k 2 )
From here we see that there is no solution iff k = 0, an infinite number of solutions iff k = 1 and else there is exactly one solution.
Feinsn

Feinsn

Beginner2022-06-23Added 8 answers

Hint:
1. No solution when:
1 k 2 = 2 2 k 1 k + 2
it holds when k = 0
2. An infinite number of solutions when:
1 k 2 = 2 2 k = 1 k + 2
it holds when k = 1
3. Exactly one solution when:
1 k 2 2 2 k
and it hold when k 1

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