1964 AMC 12 Problem 18

Attempt Problem 18 of the 1964 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1964 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

18.

Let

n

be the number of pairs of values of

b

and

c

such that

3x+by+c=0

and

cx-2y+12=0

have the same graph. Then

n

is:

0

1

2

finite but more than

2

greater than any finite number

Coincident line equations have proportional coefficient triples

Set

(c,-2,12)=k(3,b,c)

and use the first and third coordinates

For the same graph,

(c,-2,12)=k(3,b,c).

Thus

c=3k

and

12=kc=3k^2,

k=\pm2.

Each value determines one pair through

b=-\frac{2}{k}

and

c=3k.

Hence there are two pairs.

Therefore, the correct answer is C.

Problem 18 in other years