To solve for the equation that represents the direct variation between [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we are given that [tex]\( b \)[/tex] varies directly with [tex]\( a \)[/tex] and the specific values [tex]\( b = 2\frac{3}{4} \)[/tex] and [tex]\( a = -2\frac{3}{4} \)[/tex].
Let's break down what this means:
1. Understand the Relationship: In problems of direct variation, we typically know that [tex]\( b \)[/tex] varies directly with [tex]\( a \)[/tex]. This can be expressed as:
[tex]\[
b = k \cdot a
\][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
2. Substitute the Given Values: We substitute the given values into the equation to find [tex]\( k \)[/tex].
[tex]\[
b = 2\frac{3}{4} \quad \text{and} \quad a = -2\frac{3}{4}
\][/tex]
Therefore,
[tex]\[
2.75 = k \cdot (-2.75)
\][/tex]
3. Solve for [tex]\( k \)[/tex]: To find [tex]\( k \)[/tex], we solve the equation:
[tex]\[
k = \frac{2.75}{-2.75} = -1
\][/tex]
4. Write the Equation: Now that we have determined [tex]\( k \)[/tex], we substitute it back into the direct variation equation:
[tex]\[
b = -1 \cdot a
\][/tex]
Simplifying,
[tex]\[
b = -a
\][/tex]
Therefore, the equation that represents this direct variation is:
[tex]\[
b = -a
\][/tex]
To verify, here are the equations with reasons as to why they don't fit:
- [tex]\(\boldsymbol{-b=-a}\)[/tex]: This simplifies to [tex]\( b = a \)[/tex], which does not match the given values.
- [tex]\(\boldsymbol{b-a=0}\)[/tex]: This also simplifies to [tex]\( b = a \)[/tex], which again does not match the given values.
- [tex]\(\boldsymbol{b(-a)=0}\)[/tex]: This implies [tex]\( b=0 \)[/tex] or [tex]\( -a=0 \)[/tex], which is not relevant to the given values.
Hence, the correct representation is:
[tex]\(\boxed{b = -a}\)[/tex]
