If m and b are real numbers and mb>0, then the line whose equation is y=mx+b cannot contain the point

1997 AMC 12 Problem 12

Attempt Problem 12 of the 1997 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 1997 AMC 12 solutions, or check the answer key.

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12.

m

and

b

are real numbers and

mb\gt0,

then the line whose equation is

y=mx+b

cannot contain the point

(0,1997)

(0,-1997)

(19,97)

(19,-97)

(1997,0)

The condition

mb\gt0

means the slope and vertical intercept have the same sign

Substitute each point; a positive

x

-intercept forces the slope and intercept to have opposite signs

(1997,0)

were on the line, then

0=1997m+b,

b=-1997m

and

mb=-1997m^2\lt0,

a contradiction. Each other point can occur for suitable same-sign

m,b.

Thus the correct answer is E.