Question

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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 = 2-³ when a = -23. Which equation represents this
direct variation between a and b?

Answer 1

The fact that "b varies directly with a" means that there exists a constant k such that b = ka. Since "a" and "b" are located the same distance from 0 in opposite directions, we know that a and b have the same absolute value but different signs. Therefore, we can write a = -b or b = -a.

Substituting -a for b in the equation b = ka, we get:

-a = ka

Solving for k, we get:

k = -a/a = -1

Substituting k = -1 back into the equation b = ka, we get:

b = -a

Therefore, the equation that represents the direct variation between a and b is:

b = -a