To determine the equation that represents the direct variation between [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we need to understand the relationship described in the problem.
We are given that [tex]\( b \)[/tex] is located the same distance from 0 as [tex]\( a \)[/tex], but in the opposite direction. This relationship can be described mathematically as [tex]\( b \)[/tex] being the negative of [tex]\( a \)[/tex].
Consider the given example: when [tex]\( a = -2.75 \)[/tex], we have [tex]\( b = 2.75 \)[/tex].
Let's analyze each potential equation one by one to check which one fits this description:
1. [tex]\( b = -a \)[/tex]:
- If [tex]\( a = -2.75 \)[/tex], then [tex]\( b = -(-2.75) = 2.75 \)[/tex].
- This is true for the example given.
2. [tex]\( -b = -a \)[/tex]:
- If [tex]\( a = -2.75 \)[/tex], then [tex]\( -b = -(-2.75) = 2.75 \)[/tex].
- This simplifies to [tex]\( b = -2.75 \)[/tex], which is incorrect for the given example.
3. [tex]\( b - a = 0 \)[/tex]:
- If [tex]\( a = -2.75 \)[/tex], then [tex]\( b - (-2.75) = 0 \)[/tex].
- This simplifies to [tex]\( b + 2.75 = 0 \)[/tex], or [tex]\( b = -2.75 \)[/tex], which is incorrect for the given example.
4. [tex]\( b(-a) = 0 \)[/tex]:
- This equation implies that either [tex]\( b = 0 \)[/tex] or [tex]\( -a = 0 \)[/tex], neither of which are true for the given values (since [tex]\( a = -2.75 \)[/tex] and [tex]\( b = 2.75 \)[/tex]).
- Thus, this equation does not fit the relationship described.
Based on the analysis, the correct equation that represents the direct variation between [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is:
[tex]\[ b = -a \][/tex]
So, the correct answer is:
[tex]\[ b = -a \][/tex]
