We need to determine the correct equation that represents the relationship between the numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. Let's analyze each of the provided options step by step:
1. Option 1: [tex]\(b = -a\)[/tex]
If a number [tex]\(b\)[/tex] is located the same distance from 0 as another number [tex]\(a\)[/tex], but in the opposite direction, then [tex]\(b\)[/tex] should indeed be the negative of [tex]\(a\)[/tex]. For example, if [tex]\(a = 3\)[/tex], then [tex]\(b\)[/tex] should be [tex]\(-3\)[/tex], and if [tex]\(a = -2\)[/tex], then [tex]\(b\)[/tex] should be [tex]\(2\)[/tex]. This is described perfectly by the equation [tex]\(b = -a\)[/tex]. Furthermore, this matches the example given where [tex]\( b = 2\frac{3}{4}\)[/tex] when [tex]\( a = -2\frac{3}{4}\)[/tex].
2. Option 2: [tex]\(-b = -a\)[/tex]
Simplifying this, we get [tex]\(b = a\)[/tex]. This would imply that [tex]\(b\)[/tex] is exactly the same as [tex]\(a\)[/tex], which contradicts the given scenario that [tex]\(b\)[/tex] is in the opposite direction to [tex]\(a\)[/tex].
3. Option 3: [tex]\(b - a = 0\)[/tex]
Simplifying this equation, we get [tex]\(b = a\)[/tex]. Again, this implies that [tex]\(b\)[/tex] is the same as [tex]\(a\)[/tex], which is not correct because [tex]\(b\)[/tex] is supposed to be in the opposite direction to [tex]\(a\)[/tex].
4. Option 4: [tex]\(b(-a) = 0\)[/tex]
For [tex]\(b(-a) = 0\)[/tex] to hold true for all values of [tex]\(a\)[/tex], [tex]\(b\)[/tex] would always have to be [tex]\(0\)[/tex]. This does not represent a useful or general relationship between [tex]\(a\)[/tex] and [tex]\(b\)[/tex], as it doesn't apply to cases where [tex]\(a\)[/tex] or [tex]\(b\)[/tex] are non-zero.
Therefore, the correct equation that represents the direct variation and the given conditions is:
[tex]\[ \boxed{b = -a} \][/tex]
