To solve the equation [tex]\(12y + d = -19y + t\)[/tex] for [tex]\(y\)[/tex], we can follow these steps:
1. Isolate the terms involving [tex]\(y\)[/tex] on one side of the equation.
[tex]\[
12y + d = -19y + t
\][/tex]
Add [tex]\(19y\)[/tex] to both sides of the equation to move all [tex]\(y\)[/tex] terms to the left side.
[tex]\[
12y + 19y + d = t
\][/tex]
Simplify:
[tex]\[
31y + d = t
\][/tex]
2. Isolate [tex]\(y\)[/tex] by moving the constant term [tex]\(d\)[/tex] to the other side.
Subtract [tex]\(d\)[/tex] from both sides of the equation:
[tex]\[
31y = t - d
\][/tex]
3. Solve for [tex]\(y\)[/tex] by dividing by the coefficient of [tex]\(y\)[/tex], which is 31.
[tex]\[
y = \frac{t - d}{31}
\][/tex]
Given this solution, we need to determine which answer choice matches the equation:
A. [tex]\(y = 31(t - d)\)[/tex]
This implies [tex]\(y\)[/tex] is directly multiplied by 31, which is not correct.
B. [tex]\(y = \frac{t - d}{31}\)[/tex]
This is exactly what we found in our step-by-step solution.
C. [tex]\(y = \frac{t + d}{-7}\)[/tex]
This is incorrect as it implies a different expression in terms of [tex]\((t + d)\)[/tex] and a negative divisor.
D. [tex]\(y = -7(t + d)\)[/tex]
This is incorrect as it implies direct multiplication by -7 and a different expression.
Thus, the correct answer is:
[tex]\[
\boxed{B: y = \frac{t - d}{31}}
\][/tex]
