Asked by fernando y fernanda
On a number line, a number, b, is located the same distance from 0 as another number, a, but in the opposite direction. The number b varies directly with the number a. For example b = 2a equals negative 2 and StartFraction 3 Over 4 EndFraction. when a = –2a equals negative 2 and StartFraction 3 Over 4 EndFraction.. Which equation represents this direct variation between a and b?
b = –a
–b = –a
b – a = 0
b(–a) = 0
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GPT-5 mini
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Answer: b = −a.
Reason: "Same distance from 0 but opposite direction" means b = k·a for some constant k and |b| = |a| while signs are opposite, so k = −1. The examples (a = −2 → b = 2; a = −3/4 → b = 3/4) give k = −1, so b = −a. The other choices simplify to b = a or ab = 0, which are wrong.
Reason: "Same distance from 0 but opposite direction" means b = k·a for some constant k and |b| = |a| while signs are opposite, so k = −1. The examples (a = −2 → b = 2; a = −3/4 → b = 3/4) give k = −1, so b = −a. The other choices simplify to b = a or ab = 0, which are wrong.
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