find k if the system of equations has infinite solutions kx + 3y = k - 3 and 12x + ky = k?

Answer 1

If the system has an infinite number of solutions, then both equations will represent the same line. That is, one equation will be an exact multiple of the other one.

kx + 3y = k - 3

12x + ky = k

One way to make this happen is to multiply the top equation by k and the bottom equation by (k - 3), and set the equations equal to each other.

k²x + 3ky = k² - 3k

12x(k - 3) + ky(k - 3) = k² - 3k

Now pick either the coefficient of x or y from both equations and set those equal:

k² = 12(k - 3) = 12k - 36

k² - 12k + 36 = 0

(k - 6)² = 0

k = 6

Now double-check:

6x + 3y = 3

12x + 6y = 6

The second equation is twice the first, so k = 6 is correct.