Question

select the correct answer. which equation is correctly rewritten to solve for y? 12y + d = -19y + t a. y = 31(t - d) b. y = \frac{t + d}{-7} c. y = -7(t + d) d. y = \frac{t - d}{31}

Explanation:

Step1: Start with the original equation

The original equation is \(12y + d = -19y + t\). Our goal is to solve for \(y\), so we need to get all terms with \(y\) on one side and the other terms on the opposite side.

Step2: Add \(19y\) to both sides

Adding \(19y\) to both sides of the equation to get the \(y\)-terms on the left:
\(12y + 19y + d = -19y + 19y + t\)
Simplifying the left side: \(12y+19y = 31y\), and the right side: \(-19y + 19y = 0\), so we have \(31y + d = t\).

Step3: Subtract \(d\) from both sides

Subtracting \(d\) from both sides to isolate the term with \(y\):
\(31y + d - d = t - d\)
Simplifying, we get \(31y = t - d\).

Step4: Divide both sides by 31

Dividing both sides by 31 to solve for \(y\):
\(y=\frac{t - d}{31}\)
Wait, but let's check the options again. Wait, maybe I made a mistake in the sign. Wait, let's re - do the step of moving terms.
Wait, original equation: \(12y + d=-19y + t\)
Let's move \(-19y\) to the left (add \(19y\)) and \(d\) to the right (subtract \(d\)):
\(12y+19y=t - d\)
\(31y=t - d\)
\(y=\frac{t - d}{31}\), which is option D. Wait, but let's check option C: \(y = - 7(t + d)\)? No, that would be if we had a different equation. Wait, wait, maybe I messed up the original equation. Wait, the original equation is \(12y + d=-19y + t\). Let's try another approach.
Wait, maybe the user made a typo, but looking at the options:
Option C: \(y=-7(t + d)\)
Option D: \(y=\frac{t - d}{31}\)
Wait, let's re - solve the equation:
\(12y + d=-19y + t\)
Add \(19y\) to both sides: \(12y+19y + d=t\)
\(31y + d=t\)
Subtract \(d\): \(31y=t - d\)
Divide by 31: \(y=\frac{t - d}{31}\), which is option D? Wait, but let's check the options again. Wait, option C is \(y=-7(t + d)\), option D is \(y=\frac{t - d}{31}\)
Wait, maybe I made a mistake in the sign when moving terms. Wait, let's start over:
\(12y + d=-19y + t\)
Subtract \(12y\) from both sides: \(d=-19y-12y + t\)
\(d=-31y + t\)
Subtract \(t\): \(d - t=-31y\)
Multiply both sides by - 1: \(t - d = 31y\)
Then \(y=\frac{t - d}{31}\), which is option D. But wait, let's check option C: \(y=-7(t + d)\) would be if the equation was \(12y + d=-19y - t\) or something else. But based on the given equation \(12y + d=-19y + t\), the correct solution is \(y=\frac{t - d}{31}\), which is option D. Wait, but maybe the original equation was \(12y + d=-19y - t\)? No, the user wrote \(12y + d=-19y + t\).
Wait, maybe I misread the options. Let's check the options again:
A. \(y = 31(t - d)\)
B. \(y=\frac{t + d}{-7}\)
C. \(y=-7(t + d)\)
D. \(y=\frac{t - d}{31}\)
Yes, so the correct answer is D.

Answer:

D. \(y=\frac{t - d}{31}\)