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 $b=2 \frac{3}{4}$ when $a=-2 \frac{3}{4}$. Which equation represents this direct variation between $a$ and $b$?
A. $b=-a$
B. $-b=-a$
C. $b-a=0$
D. $b(-a)=0$
Answer (2)
The problem states that b is the opposite of a , meaning b = − a .
We verify this relationship with the given example.
We analyze the given options and determine that only b = − a correctly represents the relationship.
Therefore, the equation representing the direct variation between a and b is b = − a .
Explanation
Understanding the Problem We are given that a number b is located the same distance from 0 as another number a , but in the opposite direction. This means that b is the negative of a , or b = − a . We need to find the equation that represents this relationship.
Verifying the Equation We are given the example b = 2 4 3 when a = − 2 4 3 . Let's check if the equation b = − a holds true for this example. Substituting a = − 2 4 3 into the equation b = − a , we get b = − ( − 2 4 3 ) = 2 4 3 , which matches the given value of b .
Analyzing the Options Now, let's examine the given options:
b = − a : This equation directly represents the relationship described in the problem.
− b = − a : Multiplying both sides by -1, we get b = a , which is not the relationship described in the problem.
b − a = 0 : This can be rewritten as b = a , which is also not the relationship described in the problem.
b ( − a ) = 0 : This implies either b = 0 or a = 0 . This is not the general relationship described in the problem.
Final Answer Therefore, the correct equation that represents the direct variation between a and b is b = − a .
Examples
Imagine you're walking on a straight path away from your house (0). If you walk 5 steps to the right (positive direction), your friend walks 5 steps to the left (negative direction). The equation b = − a describes this situation, where a is your position and b is your friend's position. This concept is useful in physics for describing motion in opposite directions or in finance for tracking gains and losses relative to a baseline.
The correct equation representing the direct variation between a and b is b = − a , as b is the opposite of a . This was verified using the example provided. Therefore, option A is the correct choice.
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