For which values of $k$ and $m$ does the system $2x + 3y = 5$ , $4x + ky = m$ have infinitely many solutions?

k = 6

and

m = 10

Correct answerYour answer

k = 6

and

m \ne 10

Correct answerYour answer

k \ne 6

, for any

m

Correct answerYour answer

k = 3

and

m = 5

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

Infinitely many solutions need $|A| = 0$ : $2k - 12 = 0$ , so $k = 6$ .
Then the second equation must be twice the first, $4x + 6y = 10$ , so $m = 10$ . Any other $m$ makes the equations contradict: no solution.

Worth noting

$k = 6$ with $m \ne 10$ gives parallel lines that never meet, which is no solution.

Ten questions from Determinants, Adjoint & Inverse, at your own pace.

For which values of kkk and mmm does the system 2x+3y=52x + 3y = 52x+3y=5, 4x+ky=m4x + ky = m4x+ky=m have infinitely many solutions?